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Calculus 2 Lecture 6.1: The Natural Log Function — Transcript

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0:00well let's get started then on your
0:02first day of calculus 2 and like I said
0:04we're going to start with Section
0:086.1 and we're going to talk about the
0:10natural log
0:21function you know I know you guys have
0:23seen the natural log function before but
0:26it's probably a good idea to go over
0:27some of the properties because you might
0:29have forgot them uh we haven't really
0:31worked with them a lot since maybe your
0:34like pre-calculus class maybe your math
0:36C Class when you get it for the first
0:38time so we're going to start with some
0:39of the properties what you can and what
0:41you can't do with natural logarithm and
0:43with log functions in general and how
0:45that's going to help us with our
0:46derivatives and some of our integrals so
0:49let's talk about the properties the
0:50first
0:52one first one was kind of a nice one
0:54anytime you see this you loved it
0:56because it meant if you had l n of one
1:01that was always equal
1:03to say it again Z yeah it's exactly
1:05right so what what this is if you don't
1:07remember how logarithms
1:09work logarithms say a base what's the
1:13base here d e is e is the base uh for
1:17any logarithm you don't need to write
1:19this but check this out if we
1:22had this for instance this would be our
1:25base that would be our power base to the
1:28power gives you the argument of the
1:29logarithm so this would have an
1:31exponential
1:33notation like that if you forgot all
1:36about logarithms that's like a really
1:37crash course in logarithms okay so in
1:40other words when we have a natural
1:43logorithm that's based on or the base is
1:46e the natural number e e is about 2.71
1:49so on so on so on so when we have this
1:51this says Hey e to the 0o power gives
1:54you well how much is anything to the
1:56zero power that's why this works so
1:58anytime you see Ln of one we
2:01automatically know that's zero show up
2:02hands you feel okay with that so far
2:04good okay fantastic number
2:08two anytime you have and this is called
2:10the product property of logarithms in
2:12case you're wondering anytime you have a
2:14product within any logarithm
2:17including the natural logarithm you can
2:20always break up that logarithm into two
2:24different logarithms do you remember how
2:25to do it if I have log or Ln of x * y
2:30how do I split that
2:32up fantastic product plus plus product
2:35PP that's funny uh but anyway so product
2:39gives you a plus here so if we have Ln
2:41of x * y That's Ln of X Plus Ln of Y one
2:46thing to remember here these are not
2:47just oneway streets so we can either
2:50expand by a product or we can combine
2:53two logarithms by addition the only
2:55thing that you need to check on is that
2:56they have the same base now naturally
2:58our natural log always going to have the
2:59same same base so we're good to go as
3:00far as that goes you guys okay with
3:02number two yeah okay number
3:06three this one's really really similar
3:10and I'm imagining that even if you
3:11didn't know the property which you
3:13probably do you can probably figure it
3:14out if we expand
3:17multiplication by addition how are we
3:19going to SP expand division subtraction
3:22yeah fantastic which one's going to go
3:24first the Ln X or Ln y l that's right so
3:27Ln x minus Ln y again we can combine
3:31them we can combine two logarithms are
3:33being subtracted or we can expand a
3:36logarithm that has division inside of it
3:37we still okay so far y okay and last one
3:40this is my favorite one I love this
3:44one this one's called the power property
3:47says that anytime you have a logarithm
3:49with an exponent in the argument of that
3:51logarithm we can do something with that
3:53exponent what can we do with it yeah
3:56that's great so Ln of x to the r same
3:59thing as R *
4:04lnx now how we use these is to expand or
4:09combine logarithms and it depends on
4:11what situation we have but sometimes
4:13we'll be expanding sometimes we'll be
4:14combining depend on whether we're taking
4:16derivatives or integrals or doing
4:18something with that so let's practice
4:20just a couple times no calculus yet I
4:22want to make sure that you guys can
4:23expand some of these logarithms so let's
4:25start with
4:34that one Ln of 3x and you know what I
4:37want to make sure you guys know this the
4:38Ln goes in front of it Ln
4:41is it's not like multiplied by this
4:44argument it's Ln of the argument it's a
4:46it's a function of whatever your
4:48argument is here so Ln of 3x 4 all over
4:512 y^2 let's expand this as much as
4:54possible just to get the feel for how
4:56this this stuff works so um give me one
4:59idea that you might have um the minus
5:03yeah very good we we're noticing that
5:05right now we're kind of working from and
5:06you did this when you did derivatives a
5:08long time ago we're working from the
5:09outside in we don't really want to start
5:12with moving the exponents forward yet
5:14because well we got a couple of them
5:16right where would we put it doesn't
5:17really make a lot of sense so expand
5:19from the outside in the big part to the
5:20little Parts the biggest part we see the
5:22thing that combines all of this argument
5:24is division right now so let's take care
5:26of that Division and what's uh what's
5:29division changing
5:30again fantastic okay so we're going to
5:32have Ln 3x
5:364us
5:39Ln that's exactly
5:42right true or false right now it would
5:46be a great idea to move the exponents to
5:48the front no true or false false why
5:51because the exponent goes to
5:54X Y that's a great point so do you guys
5:56see what he was saying there he said
5:58because this exponent doesn't go to this
6:01whole thing it just it's just attributed
6:03that X I can't move forward yet now if
6:06it had been like this would I move it
6:08forward that'd be fine but if it's not
6:10you got to be real careful on that do
6:11you guys see the simple mistake that
6:13that can really really cause a problem
6:15right so watch the small stuff
6:17here all right well if we can't move
6:19that forward what's something that we
6:21can do with that logarithm what do you
6:24think how would I break them up
6:30to yeah you're exactly right because
6:32that's a product in there I can
6:34certainly do
6:38that let's do the same thing over
6:46here okay I want you think carefully on
6:48what I've just WR wrote on the board am
6:50I right or am I wrong wrong some over
6:54here why am I what did I do here
6:58negative need the go to
7:00fantastic so if we're subtracting this
7:02Ln remember if you expand this L this
7:05still means this right it still needs to
7:07be subtracted so either you're going to
7:09draw those brackets or you're going to
7:11distribute that negative that minus sign
7:14you guys with me on that one again be
7:15careful the small stuff so we're going
7:17to do a couple steps right now there's
7:19one thing I want you to do to expand
7:20this one more time what what can we do
7:22now with our
7:24logarithms Lou now you can do like the
7:26exponent to the front yeah that's
7:27absolutely right and then I'm going to
7:29distribute that minus so we'll have Ln 3
7:33+ 4 Ln
7:35xus ln2 nothing we can do there but here
7:38we'll have minus 2 Ln y this when we get
7:41to this point there's literally nothing
7:43else we can do to expand this make it
7:45any longer than than what we have right
7:46now by show of hands do you feel okay
7:48with expanding our logarithms good with
7:51it any questions at all what you're
7:54going to find out is that because we can
7:56do all this what's going to be easier to
7:58take a derivative of
8:00this argument or that
8:01argument that's way easier that's a
8:04constant forg good to sake derivative
8:05constants are hey that's fantastic this
8:09going to be a piece of cake this zero
8:11this going to be a piece of cake so
8:12we're we're getting there but by
8:13expanding logarithms we make our
8:15derivatives and our integrals a whole
8:17lot easier when we're dealing with Ln
8:19okie dokie
8:23okay tell you what we'll try um we'll
8:26try one more together and I want to see
8:28what you can do on one of
8:32them just keep something in mind too
8:34these properties they're the only
8:36properties that we have for logorithm as
8:37far as expanding combining one big thing
8:39people do in math is they like to make
8:42up their own math their own properties
8:45look if they weren't listed on the board
8:47you can't do them so be real careful
8:48about what you can expand and what you
8:50can combine so don't make up your own
8:52stuff just follow the things that they
8:53given you so let's see what we can do
8:56with this again we'll do uh one more
8:58together and one on your own
9:07so Ln x^2 + 1 all over the < TK of
9:11X tell me someone on the left hand side
9:14of the room you guys over here if I want
9:16to expand this all the way where might I
9:21start Division I would say change the X
9:24to the like theot to the 12 first I
9:27could do that sure uh I can do that now
9:29or I can do it later do you guys see
9:30what he was talking about the square
9:31root we know that's 1/2 power so we're
9:33going to do that ultimately either now
9:35or later doesn't really matter when as
9:36far as expanding goes where would I want
9:38to start would I want to start moving
9:40exponents forward would I want to start
9:41separating products or separating uh
9:44quotients which one you do the
9:45subtraction thing Okay cool so we know
9:48that this is the same thing
9:52as well since that's a quotient I know
9:56I'm going to have subtract square otk of
9:58x
9:59one more thing just a little side note
10:01for you guys whenever you're expanding
10:03sometimes you'll have to put some
10:04parentheses in the argument of your
10:05logarithm especially like right here
10:07this is really ambiguous it doesn't
10:09really say whether I have Ln of x^2 + 1
10:12or Ln of x^2 + 1 you want to make sure
10:15you show that um don't leave the little
10:17stuff hanging out especially in calculus
10:192 because you get a l the the way that
10:22you write problems is important it's not
10:24meaningless like when you're doing math
10:2680 you almost never get wrong right you
10:27just write anything you want uh but here
10:30at this level you really need to be
10:31careful about about how you're writing
10:33things so make sure you're you're
10:34accurate now I like the one2 we're going
10:36to change in just a moment what I want
10:37to know right here can I split that up
10:39anymore can I expand that any longer no
10:42that's great no we can't do it at all we
10:44know that if it was a product we could
10:46expand it but if it's plus there's
10:48nothing that says we can expand plus so
10:50if we can't expand the plus we can't
10:51move the two forward we're basically
10:53done here the only thing we can do and
10:56this was already said
11:01that is actually a
11:04power and that can move
11:14forward that's about as good we can get
11:16uh we can't split that up for sure we've
11:18already moved our power forward and
11:20that's expanded I want you to try one
11:21your I'm going give you a hard one uh if
11:22you can do this one you know what you're
11:24doing
11:47booah I think it deserves a booya let's
11:48go for it show me what you got remember
11:50that what we're doing work from the
11:51outside in go for the big Parts first
11:54and split off everything that you can
11:56follow only the properties that I've
11:58given you we can separate products we
12:00can separate quotients we can move
12:01powers to the front so do the best you
12:24can by the way when I have you do
12:26something like this I'll typically walk
12:27around the room if you have questions by
12:29all means please ask me that's is what
12:31this is for
13:06yeah to
13:47when we have a a cube root what's the
13:50power of that cube root cool so a square
13:54root because we had a square root we
13:56have 1 over two we got power over root
13:58basically the power of the inside over
14:00the root so when we have a cube root
14:02don't be putting crazy things like 2/3
14:04or anything weird it's just 1/3 unless
14:06you have a power like this if we had a
14:09cube root of X2 that's when you would
14:11get a 2/3 power or something okay it's
14:14power over root so we have hey no
14:17problem the first thing I saw I've got
14:19division here this is containing this
14:20whole argument so I'm going to separate
14:22that division by a subtraction from the
14:23logarithm I'm also going to change that
14:25power to 1/3 power and that says I can
14:28move that out front next thing we're
14:29looking for is any remaining products or
14:32quotients in here um do you see any
14:34other products in here how many one or
14:37two good you know some some people who
14:41just don't like trigonometry they'll say
14:43oh yeah you got one right here is this a
14:44product no no this is actually the
14:46argument of a sign so this is the really
14:48the angle of that sign function so we
14:51are going to separate this product but
14:53we can't separate this it's not a
14:56product so we'll do Ln of x 5
14:59we'll do a plus sign because we got
15:01product plus Ln of sin
15:062X and we'll do
15:08minus Ln of x to
15:134 + 2 to the 1/3 and lastly anywhere you
15:17have a exponent of an
15:19argument are we okay anytime we have an
15:23exponent of an argument we're going to
15:24move it up front so we have 5 Ln X
15:30the two can I move the two
15:33forward what do you think what does this
15:36mean when I
15:53have what does that mean X's to the 4
15:56power is it the X is it the function
16:05function that's literally what that
16:06means so whenever you see a sign squared
16:09like this this is not being squared if
16:11it was I would have it here uh don't
16:14change your work please but
16:15watch if this was being squared I'd have
16:18it here and I'd have that as the AR
16:21that's how that would
16:23look if I don't want to say that if I
16:25want to say the functions being squared
16:27I know it's a little weird uh but people
16:29have gotten the habit of just writing it
16:30right there that's the power of the
16:32entire thing okay the entire function of
16:34sign that's being squared right now so
16:37we have this that's really a power
16:39that's an exponent we can move that to
16:40the
16:46front of course our
16:521/3 okay we've broken it down can we go
16:54any further do you see any products any
16:56quotients or any powers that we can move
16:58move no let's check
17:03no this well that's connected we can't
17:05separate addition we can't move that
17:06power forward we're completely done show
17:08up hands if you got that on your own
17:10that's really good you guys know what
17:11you're doing then you know just like we
17:13can expand our logarithms we can also
17:16combine them I just want to practice
17:17real quick doing one or two of those
17:20then we'll start talking about how we
17:21take derivative of these things
17:35you guys seem like really smart people
17:36so we'll cut right to the chase we'll do
17:37hardw
17:59so what I want to do here I want to
18:02combine these logarithms to see if we
18:04can get as as few logarithms as possible
18:07in our case we're trying to get one so
18:09one logarithm so let's take a look at it
18:12you know what you might have noticed is
18:14that we typically deal with our
18:17exponents
18:18last when we are expanding so here we're
18:23going to probably have to deal with some
18:24of these guys first otherwise we can't
18:26combine them does that make sense in
18:28order to find our logarithms when you
18:29when you notice those properties earlier
18:31it was log X Plus log y gives you log of
18:35x y or Ln X Plus Ln y there's no numbers
18:38up front in other words we have to move
18:40our numbers up front our coefficients to
18:41exponents first so tell me one thing
18:43you're going to do here should I deal
18:45with the 2/3
18:47first that one we can't okay that's
18:49that's out there this is going to be the
18:50last thing we do should I deal with the
18:52five first yeah that's got to move up
18:55and the 1 half has to move up and the
18:57four has to move up so we'll leave that
19:002/3 it says order operations anyway
19:03really doesn't it it says 2/3 is outside
19:05deal with this stuff first put this all
19:07together then do the
19:092/3 so inside of our our bracket here we
19:12got an Ln of x to the 5th we've got a
19:19Ln x^2 + 1 to the
19:2312 we got a minus
19:29Ln x - 2 all 4
19:34power and I'll finish the
19:36FR okay let's keep on on working uh
19:39right side people you guys what's the
19:40next thing you might want to
19:42do say
19:44what M sure you can actually do both at
19:47the same time if you really know what
19:48you're doing if you know this is going
19:50to be combined by multiplication and
19:52this one's going to be combined by
19:53division we're just going to have this
19:56times this all over that just do me one
19:59favor if you're going to combine these
20:01things please don't write different
20:03logarithms it becomes one logarithm it
20:06never becomes Ln over Ln that's not the
20:09idea we're getting Ln of this argument
20:12times this argument divided by that
20:13argument all in one little
20:16piece so the 2/3 is still hanging on we
20:20got
20:21Ln well we got X to 5th we know plus is
20:25going to give us a product
20:30tell me one other thing you could do
20:31with that one half power what could you
20:33change it to if you wanted to all
20:37over you were like my little math robots
20:39there it was
20:40awesome square
20:44root there we
20:47go okay as if it's not ugly enough we
20:51can do one more thing what can we do
20:53now now that we look at that that's a
20:55coefficient right in fact I don't even
20:57really need these brackets if I didn't
20:59want them the 2/3 could go right there
21:02if that's a coefficient of a logarithm I
21:04can move that up and becomes a power now
21:07so this our very last step is
21:10Ln x to the 5th whole bunch of nasty
21:18crap and then the whole thing now with
21:21the bracket inside you do need the
21:22bracket inside to the 2/3
21:26power okay that's about as inp as I want
21:28to get do you guys feel comfortable on
21:30expanding and combining logarithms
21:33probably some good practice right
21:34because you haven't seen it in a
21:35while could we have and that first step
21:38it would have been okay to just
21:39distribute the 23 you just would ended
21:42up with smaller exponents at the end
21:44absolutely uh you could do here and here
21:47and here and then move those up uh sorry
21:50uh here here and here and here and move
21:53those Powers up you'll notice that you
21:54get the same exact thing because well
21:57what's interesting is that that when you
21:59distribute you multiply don't you you
22:01multiply and you multiply and you
22:03multiply what do you do when you take a
22:05power to a power
22:07you how about that it's the same stuff
22:09so even if you take a power to a power
22:11you're still multiplying you get 10/3
22:13same thing would happen here you get
22:1510/3 okay so either way you go you'd be
22:17fine I like this way because this is all
22:19to one power that's just uh preferential
22:22though good question thank you for that
22:24one any other questions at
22:26all are you ready to move on to finally
22:28get dis
22:37calculus okay now that we've covered
22:40that we will use it trust me we'll use
22:41it a
22:43lot I want to refresh your memory on one
22:46more thing and then we're going to start
22:47talking about some integrals and get on
22:49to what in the world we're going to be
22:50doing with uh all this junk so the last
22:53thing we got to cover do you remember
22:55the graph
23:02of this
23:10function okay do you remember the graph
23:12of this
23:18function I see a lot of already let's
23:22talk about it when you take a positive
23:25number which this is uh e is
23:27approximately 2.7 you take a positive
23:29number to any power what are you gonna
23:32get
23:33out you're gonna get out a positive
23:35number that's right can you ever get out
23:40zero can you ever get out
23:43zero no so what automatically know that
23:46this function is not going to cross the
23:48x-axis it's always going to be above the
23:50xaxis does that make sense to you okay
23:52so uh how about this what if you were to
23:54plug in zero here how much would you get
23:57so we know it's going to across the y
23:59AIS at
24:02precisely
24:0401 you said that it's always going to be
24:07above the x-axis because it's a positive
24:10number being raised to a power it has to
24:11be above here you're never going to
24:13cross this line does that make sense uh
24:16now when I plug in big numbers does this
24:18thing get really big or really
24:20small small big numbers Take 2 to 4th 2
24:25to the e8th two this is about two this
24:27going to biger or smaller bigger bigger
24:30way bigger way
24:35faster what's going to happen over
24:38here because when you have e to the -2
24:42which is what the these are that's the
24:43same thing as 1 over e s does that make
24:45sense when you have a negative exponent
24:47you put on the dener so as that power
24:50gets smaller and smaller more and more
24:51negative that denominator gets bigger
24:53and bigger and bigger and bigger one
24:55over a really big number tends to be
24:57close to zero is it never going to touch
24:59zero no but it's going to look like
25:03this that's the graph of e to the
25:14X tell me something somebody out there
25:18what's the relationship
25:20between this function and that function
25:24inverse very good they're inverses what
25:26inverses are graphically are a
25:29reflection across the yal X line so if
25:32we want to find out that graph it's
25:34really stemming from this graph and
25:36other words if I take
25:39this this diagonal right here and I just
25:42flip it right over that's going to give
25:44me Ln
25:50X so tell me if
25:5301 is on my e to the X function if 01 is
25:58on my e to X function what has to be on
26:01my Ln X function good because inverses
26:04just flip the X and the
26:09Y did you know that about inverses that
26:12inverses really just flip the X and the
26:14Y it's a reflection across yal X so it's
26:17just flipping those coordinates really
26:19so if we have 0 1 and inverse would have
26:211 0 if this is astoic
26:26tox this is Asm prodct to
26:32Y which means our graph looks about like
26:37that that's the graph of Ln X you kind
26:40of need to know the graph of what we're
26:41working with because the properties are
26:43going to make a whole lot more sense to
26:45you uh for instance the very first
26:47property said hey when you plug in one
26:49you're going to get zero hello when you
26:52plug in one you get zero remember that
26:54Ln of one is always zero it's on the
26:57graph it's right there can you plug in
26:59zero to
27:00Ln do you have a
27:02calculator have you ever tried to plug
27:04in zero to Ln press Ln Zer if you have a
27:06calculator right now try
27:18it try do you have an error I
27:23got did you get if you tried it if you
27:25press
27:26ln0 hopefully if your calculator's not
27:28broken it's going to give you error what
27:32are you trying to kill me that's what
27:34it's saying you're breaking its heart
27:36really is what you're doing uh because
27:38you're supposed to know this about n can
27:41you plug in
27:43zero it's undefined it's undefined can
27:46you plug in a negative to Ln do you
27:47think it's undefined there's nothing
27:49there so when you realize what the graph
27:52looks like you kind of understand you
27:54understand a little bit better about why
27:55you can't plug in why you can't have Ln
27:57of three it doesn't make any sense you
28:00can't have Ln of Zer it doesn't make any
28:02sense Ln of one gives you zero it's on
28:05the graph so when you get a better
28:06picture of what the graph looks like Len
28:08makes a little bit more sense she H feel
28:09okay with the graphs of these things did
28:10I do it well enough for you understand
28:12where it came from fantastic now let's
28:15see how good you are how much you
28:18remember about one of our favorite
28:20things in the
28:23[Applause]
28:26world you ready
28:34R the bat man I'm going for
28:38it holy cow oh dear you didn't think it
28:41was going to be this fast did you don't
28:44wor you should know this stuff right you
28:45should know the integrals integral oh my
28:47gosh integral is the same thing as what
28:49we call an anti-derivative basically the
28:51not basically it is the inverse of a
28:52dtive just going backwards do you
28:54remember how to take the integral of
28:56something like that where we have x to
28:57the 3
28:58DX which says in with respect to X
29:01changing
29:02add add something add one very good so
29:07the power rule backwards said well hey
29:09if you take a derivative and you
29:10subtract one you move forward and you
29:12subtract one here we're going to add one
29:15and then divide so if we did x to the
29:185th as a
29:21derivative and that said move this
29:24forward then subtract one we're going to
29:27do complet completely the opposite thing
29:29now we're going to add
29:32one and divide by that new exponent so
29:36this said multiply by the old exponent
29:38then subtract one this one says add one
29:41to the exponent then divide by that new
29:43exponent that's exactly how you take an
29:45integral and I'm missing one thing what
29:47am I missing why do we have a plus
29:51c what we're doing here is we're finding
29:53a family of curves that work you see
29:59that plus C when you take a derivative
30:00of any constant is going to go to zero
30:02so we have a whole family of Curves at
30:04work the only time you don't have a
30:05family of Curves with integral is if you
30:06have either a definite integral or you
30:09have an initial condition uh which we'll
30:10talk about
30:12later you okay with that one basic basic
30:16okay let's move on just a little bit
30:18what would happen if I
30:23had a constant inside of my integral
30:28always move out a constant provided it's
30:29being multiplied not added not
30:31subtracted but multiplied by whatever my
30:34function is so sure we can move that
30:35outside we can go four times the
30:37integral just like derivatives we could
30:38do this x 5th DX uh tell me left side
30:42people only just you guys tell me the
30:44integral of x to the 5th
30:49please cool okay however you want to
30:51write that's fine so we'd have four just
30:53don't forget the four x 6 / 6 + C and
30:57what we going to do with that simplify
31:01absolutely however you want to write it
31:03if you want to do the
31:052/3 x 6 + C or 2 x 6 over 3 I don't care
31:11whatever you want to do there
31:13fine still okay so okay stretch your
31:16brain just a little bit think back to
31:18this
31:26one oh dear
31:29now I want you to really think about
31:31what happens with derivatives before you
31:33answer this unless it's just burning in
31:35your memory you know it but please think
31:37before you answer this
31:39question integral of sin x DX
31:45integral cosine or negative
31:48cosine why isn't it positive cosine
31:51netive sign good the derivative of
31:53cosine is negative sign so if I'm trying
31:55to get a positive sign well the
31:58appropriate anti-derivative here is
31:59negative cine
32:01X watching too much
32:04scrubs Dr
32:09Cox bless bless you thank you yeah plus
32:12see
32:14exactly last one
32:33let's try that one up for size what are
32:34you going to do with that
32:36problem uh the qu quo rule works for
32:40derivatives but like where he heads
32:42that substitution oh I love substitution
32:45I love substitution if oh my gosh your
32:47teacher should have taught you this it
32:48wasn't me uh if the derivative of what
32:51you're trying to substitute is actually
32:53in the integral so find a piece of there
32:56take a derivative
32:58do you see that piece in the
33:01integral you do I hope
33:03not that gives you 4X to the 3 right I
33:06don't care about the four but do you see
33:08the X to the 3 I don't either plus is on
33:11the denominator so that would be a
33:12problem
33:13too think it easier SE top so what do
33:17you mean separated like x 4 x
33:222us that let's do that so this is kind
33:25of a nice trick anytime you have a
33:27single term denominator you can always
33:29separate it you always make different
33:30fractions out of that so we could
33:35do x 4x^2 - 1x^2 yeah
33:41absolutely that's fantastic tell me the
33:44next thing we're going to do someone
33:45over here on the left hand side what now
33:47simplify simplify one thing for me what
33:49do you get would be X2 on top m-
33:54One X x s but you can get X to the yeah
34:02perfect go ahead and do the the integral
34:05forming on your own see if you can get
34:07that
34:32remember that while you can split this
34:35up into integral minus integral you
34:37don't have to you can just take it piece
34:40by piece so take the integral of this
34:41piece minus the integral of this piece
34:44of course you can do negatives you can
34:45change the minus to a plus you can do it
34:47just piece by piece when you have
34:48multiple terms in there you don't have
34:49to separate it
34:53all uh by the way just a little bit of
34:55of notation for you when you do the
34:58integral and you get X 3r over 3 or 1/3
35:02X3 you guys got that right I know this
35:05because I I saw a lot of you in the math
35:07lab some of you when you do
35:10integrals by the way what's this
35:12become one good you add one to it and
35:15you divide by whatever the new exponent
35:17is did you all get that far okay some of
35:19you like to do this some of you like to
35:21take integrals and go oh yeah I just
35:23took an integral what that symbol means
35:25is that you're supposed to take the
35:27integral whatever the argument is next
35:29if you've already done the integral this
35:31does not go here okay you've already
35:32done that so erase
35:34that you you want to stand about the
35:36notation here this is telling you what
35:38you're supposed to do this says take an
35:40integral evaluate it from 1 to two this
35:42says take an integral haven't you
35:45already done that if you want to show
35:47the evaluation the appropriate way to
35:48show it is to put it over here from one
35:51to two that's where you put that you go
35:53okay I've done the integral it's a
35:55definite integral now I'm going to
35:55evaluate from 1 to two of course we want
35:58to make it a little bit nicer looking
36:00before we do that we'll do X 3r over
36:053 plus one do you guys see where I'm
36:08getting the plus one overx from good
36:11cool and then we would evaluate it go
36:13ahead and evaluate it remember that when
36:16you evaluate we plug in this number
36:18first we figure out what that answer is
36:20and we subtract after we plug in that
36:22number so plug in the two figure out
36:24what you get subtract after you plug in
36:27one
36:47did you get
36:48it what' you get 116 116 double
36:54check 11 116 116
37:01we're not going to spend a whole lot of
37:03time in here waiting for this I know you
37:05guys can do this I just want to make
37:06sure you see some finer points of doing
37:08integrals that way you don't get stuck
37:09on some of these easier ones okay don't
37:11don't make it harder than it is that's a
37:12lot with this classes about don't make
37:14it hard it is so fans be okay with our
37:17integrals okay I have one more for you
37:30give me
37:32ideas the top
37:37okay that looks pretty painless let's
37:40try that so every other time we had this
37:42we did uh let's see where were we add
37:45one divide by the new exponent are you
37:48listening add oneide by add one divide
37:50by the new exponent let's let's do that
37:52let's add one to this uh what do you get
37:55x to the well that's fine we' get one
37:58divid by that's not fine that's not fine
38:02at all that that's pretty nasty so what
38:04in the world are we going to do oh my
38:07goodness let's give up calculus 2 is
38:09over let's just
38:11go well this isn't going to work this is
38:13not good this is not good at all well
38:16let's think a little bit more about this
38:18um first things first do you know what
38:20the graph of 1x actually looks
38:24like think about it for a second do you
38:26remember what the gra of one/ X actually
38:28looks
38:30like approaches zero it does approach
38:33zero and it approaches Infinity it's
38:36like this little actually got the wrong
38:39sound effect my bad sort of thing what
38:43I'm going to do and you'll see Y in a
38:44minute I'm going to graph 1 over T
38:46you'll notice that 1 over T has a dummy
38:47variable in place of x uh you'll see Y
38:49in a second this
38:52graph if this is T and this is
38:56y looks about like that the graph looks
38:59like as I get closer to zero my number
39:01gets really really big positively as I
39:03get closer to Infinity my number gets
39:05really really small but still positive
39:07you with me okay true or false that
39:10function is continuous anything bigger
39:12than
39:13zero absolutely there's no holes there's
39:16no gaps um true or false it's
39:18differentiable anywhere past zero are
39:22there any sharp points no so this is
39:25continuous and differentiable well what
39:28that means by the fundamental theorem of
39:31calculus I'll write this out for
39:35you
39:38because 1 / T is
39:42[Applause]
39:49continuous on the interval that we're
39:51trying to talk about here
39:57by the I'm abbreviate by the fundamental
40:00theorem of calculus part
40:04one faki I made that up by faki by
40:09fundament sounds kind of cool though I
40:11think I think it'll catch on my YouTube
40:13people make sure it catches on the
40:15talking by the fundamental theor of
40:16calculus part one this
40:19thing oh sorry we must be able to find
40:22some differentiable function that gives
40:23us this thing
40:35for
41:03because this functions continuous on our
41:06domain we're looking at by the
41:08fundamental theorem of calculus we
41:10absolutely have to find some
41:12differentiable function we're going to
41:13call it capital F ofx big F ofx such
41:15this is true such
41:19that F ofx big f ofx equals the
41:25integral of our function from some
41:29defined point to X that's got to be the
41:33case if you don't believe me look up the
41:34fundamental theorem of calculus part one
41:37that's exactly what it states so this is
41:38just by definition I'm not making
41:39anything up here okay uh this says
41:42that this integral has to exist it's got
41:45to exist because it's continuous and
41:47fundamental theorem calculus part one
41:48which is proven says it exists it's got
41:50to be there also says this is just the
41:53the corollary
41:54that if I take a derivative of it
42:00that would be again since these things
42:02have to be equal it would be a
42:03derivative of this thing this you have
42:06seen before this is what you were taught
42:09in calculus one about the fundamental
42:11theorem of calculus it says when you
42:12take a derivative of an integral and you
42:14go from a defined point to x what this
42:17is is just 1
42:20/x do you remember that yeah which is
42:23the function in terms of x
42:28these are the easy ones you had in the
42:29test where it's like oh I got a whole
42:31bunch of T's D I just put an X
42:32everywhere I see T do you remember that
42:34one it's basically what it said and now
42:36you know a little bit further what this
42:37is talking about is that if your
42:39function is
42:41continuous on this interal you have to
42:44be able to do this you have to be able
42:45to take the integral of that function uh
42:47and it it's defined it's a
42:49differentiable function and when you
42:50take of that function it gives you back
42:51your original function so uh basically
42:55it says there's got to be some function
42:57that gives me little F ofx when I take a
42:59derivative of it and the integral of
43:02little F ofx actually exists that's the
43:04basic idea here now what we do to make
43:06it just a little bit easier um we
43:09typically Define a to be one it really
43:12doesn't matter as long as it's bigger
43:13than zero and this thing works out
43:15pretty nicely so here's a couple things
43:17I want you I want to point out to you in
43:18the last five minutes
43:28you know what I'll write your
43:30too it turns
43:34out that the following things are true
44:01sorry my bad that should be a t there
44:03change your notes that needs to be a t
44:05I'm sorry you get the X when you
44:06evaluate from a to X I
44:13bet so first thing it turns out that lnx
44:17is actually equal to the
44:19integral of 1/ T from 1 to X that's a
44:24true statement it's by definition so
44:25that's a definition that we have
44:29also check this out this is where the
44:31fundamental theorem of calculus part
44:32one's going to come in watch very
44:34carefully okay did you know that you can
44:36do anything you want to to an equation
44:38as long as you do it to both sides yeah
44:39including take
44:41derivatives which is cool let's check
44:43this out if I do you believe me that
44:45this is true you have to darn you
44:47because it's a definition it's true if I
44:50take a der because it said so if I take
44:52a derivative of both
44:54sides check out what happens over here
44:57here if I take a derivative of Ln X I
45:00have to take a
45:02derivative of this nasty
45:08crap that's a little cramed sorry a
45:12little
45:14better
45:16derivative l
45:20x equals derivative
45:31do you see this piece anywhere else on
45:33the board that we' talked
45:36about what's it
45:42equal okay check it out by fundamental
45:45theorem of calculus part one we know
45:47that this is true by definition we know
45:50that this is true so by the transitive
45:52property if this we can take a
45:54derivative of both sides this equals
45:56this this part equals
46:06this oh you know what I'm going to do it
46:08I'll write over here so you have more
46:11room did I lose you do you guys have any
46:13questions on that I don't want to lose
46:14you on that one it's kind of a cool
46:15little proof um I hope it makes
46:20sense so here's what you need to know
46:22this is a really long way to say here's
46:25what it is so here's what it is is
46:27here's what you got to know the first
46:29thing yes we can now take integrals of 1
46:32/x check it out if the
46:35derivative of Ln X = 1X that means the
46:39integral of 1x = Ln X the only thing
46:42that we have to be careful of is that we
46:44have to have an absolute value around
46:46the argument of Ln naturally we can't
46:49have an argument that's negative
46:50remember the picture remember the graph
46:51we can't have negatives in there uh so
46:54here's what we say we say this equals
46:57the Ln of the absolute value of x of
46:59course plus C that's number one thing so
47:02when you're taking the integral and you
47:04come across 1/x and you go holy crap if
47:07I add if I add one to the exponent I get
47:09zero over Z I get one over zero we can't
47:12do that well now you have something for
47:14that we've proven it over here when you
47:16take the integral of 1/x DX you get Ln
47:18absolute value of x plus C are you guys
47:20okay with that one did you get the the
47:21reason behind it if the derivative of l
47:23xal x almost the RSE is true the
47:25integral One X giv you Ln of absolute
47:28value of
47:30x if you
47:33have a definite enru the same thing is
47:38true we get Ln
47:43X from A to B if you're wondering well
47:47wait a minute Mr Leonard uh wait a
47:50second I thought you said that you had
47:52to have absolute value here check out
47:54what would have to happen for this to be
47:56integrable function you'd have to have
47:59something that doesn't go across zero do
48:01you see what I'm talking about you'd
48:02have to either have both positive both
48:04negative we're talking about both
48:06positive here so if you have both
48:08positive you're never going to have that
48:09negative argument inside the logarithm
48:11do you see what I'm talking about so
48:12we're fine as far as that goes the
48:14second
48:16part if we take a
48:23derivative of Ln absolute value of x we
48:26are going to get
48:291X say right here derivative of Ln X =
48:331X we have the absolute value so it
48:35looks a little bit more like this um
48:37really you're not going to see a whole
48:39lot of the absolute value unless you can
48:41have a function which goes negative like
48:43cosine or something okay you're going
48:44see that little bit so really this or
48:54this it doesn't really matter it's kind
48:56of all this is saying is that hey your
48:58arguments got to be positive so you make
49:01it positive should fans feel okay with
49:03with that one okay tell you what I know
49:05we haven't done a whole lot of examples
49:06we are going to get to that next
49:11time so today we're going to do a whole
49:13lot of examples with derivative of Ln
49:15last time what we learned was the
49:17expansion and the combination of natural
49:20logs we also learned a little bit of
49:22calculus behind it we learned that if we
49:24take the derivative of Ln X it's 1 /x
49:26we're going to do the integrals in just
49:27a bit but I want you to practice some of
49:29these derivatives right now before we
49:31get to it though uh there's one thing I
49:33want you to recall from your Calculus
49:34one class it was this it's if you take a
49:39derivative of some function with another
49:42function of X inside of it we called
49:44that do you remember what that was
49:46called chain rule yeah that's right this
49:47chain rule so if you remember the chain
49:50rule what the chain rule said to do was
49:53take the derivative of the outside
49:55function in this case F I call it the
49:57outside function it's just like the big
49:58function what's containing everything
50:00take the derivative of f leave the
50:02inside alone and then multiply by the
50:05derivative of the inside do you remember
50:06doing that so in in this case it would
50:08look like this it would be F Prime of G
50:12ofx youd leave that alone and then you'd
50:14multiply by the
50:16derivative of G
50:19ofx or you just say G Prime of x if you
50:22want I'm kind of mixing notation here uh
50:24one quick example if you want a really
50:26quick
50:27like little sub example here if you did
50:30like the derivative of x^2 +
50:351 to the thir power if I asked you for
50:38that derivative what we would do here
50:40I'm not going to ask you to to help me
50:42out on this one because you really you
50:43should know it I'm just going to go
50:44really quick through it this would be a
50:45chain rule uh some people call it the
50:47general power rule it says the big
50:49function is something a function to the
50:51third power that's the big function here
50:53do you see how it contains everything
50:55says you bring down the three like you
50:56normally would you leave the inside
50:58alone whatever's here goes here you
51:01subtract one from the power just like
51:03you normally would notice three subtract
51:06one becomes two but then you multiply by
51:08the derivative of whatever the inside
51:10function is our G of X in this case
51:12would be x^2 + 1 you guys recall that
51:16show up hands if you remember that okay
51:17and then of course you do the derivative
51:18you get 2x you pull it out front and
51:20you'd be done with our derivative I'm
51:22not going to finish that off for you but
51:23that's the idea we can do the same thing
51:26with lnx I want you to consider what's
51:28going on here if we
51:33have a derivative of Ln with some
51:37function inside of
51:42it in our case right now this F the
51:47function f is just Ln so the function Ln
51:50it takes the place of the function f
51:52right here does that make sense to you L
51:53natural log is a function so it's taking
51:56the place of f if we have any sort of
51:58function of X inside of that what we're
52:00going to do here we're going to do the
52:01chain rule the chain rule says all right
52:04well the first thing you need to
52:06consider is take the derivative of Ln
52:10what's the Der of Ln what's the Der of
52:11lnx say to do you put one over so you
52:14put one over whatever the function is in
52:16this case this is really easy because we
52:18had a function of X it was just well the
52:20Der of X is one really what you're doing
52:22here is the chain rule it says you're
52:24putting one over whatever's here so 1/x
52:28and if you think about it it's kind of
52:29weird but if you think about it if you
52:32multiply by the derivative of the inside
52:36here which is X what's a derivative of x
52:38y so you it's not hopefully it's not
52:40zero because the whole thing would go
52:41zero it is is one so we don't show that
52:45because a derivative of x is just one
52:47but that's really what you're doing here
52:49okay you're doing one over the inside
52:52times the derivative of the inside the
52:54same thing works here we're going to do
52:57one over the
53:00inside times the derivative of the
53:05inside what I want to be sure is that
53:07you guys see both Ln at work and that
53:10you see both the chain rule at work do
53:11you guys see both those things that work
53:13right now the 1 over G of X that's the
53:15same thing we're doing here this is 1 /x
53:17it just so happens we're calling the
53:19function inside G of X so one over the
53:22inside one over the inside times the
53:24derivative of the inside times the
53:25derivative of the inside that's the
53:28chain rule in respect with respect to Ln
53:30show hands you feel okay with that one
53:32okay we're going to practice that in a
53:33minute for those of you who had a
53:34different type of uh notation for chain
53:37Rule and give it to you this way as well
53:40it means exactly the same
53:42thing it's just some people like to see
53:45the U here where U is a function of X so
53:48if you have Ln of U where you is a
53:50function of X it looks just a little bit
53:52different I hope you understand it means
53:54the same thing you put one over U
53:57you can see it's the same exact thing
53:59times the derivative of U with respect
54:02to X DX same same idea just a little bit
54:05of different notation what do you say we
54:07practice some examples right now would
54:08you like that okay let's go for it uh I
54:11promise you these really aren't that
54:13hard if you keep in mind this is a chain
54:14rule what a lot of people do here is
54:17they'll they'll forget to do this part
54:19just like a lot of you in your Calculus
54:21one class when you first learned it some
54:23well maybe not a lot of you but some of
54:24you forgot to do this a lot you probably
54:27did especially with like s of 5x you
54:29know where 5x is inside I know when I
54:31taught calcul this one a lot of people
54:33forget that that's what's not cosine of
54:355x it's 5 cosine 5x because you have to
54:37take the derivative of the inside same
54:39thing happens here don't forget that
54:42when we're doing logarithms you still
54:43have to take the derivative of this
54:45piece it's kind of like two derivatives
54:46derivative of Ln is here derivative
54:49inside is here let's practice it
54:54[Music]
55:04[Music]
55:12okay so we got a derivative we got Ln
55:14you guys seeing what I'm talking about a
55:15function within within a function here
55:18we got a function of X inside the
55:19function Ln so the first thing that our
55:22chain rule says to do is all right take
55:24the derivative of the outside function
55:26in this this case the Ln function
55:27contains everything so if we take the
55:29derivative of Ln it says you're going to
55:31put one over I don't even care what's
55:33here you put one over whatever is on the
55:34inside so we're going to have 1 over
55:373x^2 - 1 looks a whole lot like this
55:40idea right now you're taking a
55:41derivative of the power but you're
55:42leaving the inside alone taking a
55:44derivative of Ln you're leaving the
55:45inside alone now what do we do leave it
55:48keep going keep going by multiplying by
55:52what sure I would like to see this work
55:55right now you guys get all advanced and
55:57you're in forc or whatever do whatever
55:59you want okay but right now I want to
56:00see you doing this that way if you make
56:02a mistake I can see where it's coming
56:03from so I want to see your work on these
56:05problems I want to see the one over this
56:07I want to see times a derivative some
56:09form you can use prime if you want I
56:11prefer the DDX because when I teach
56:13calculus I tell people hey follow the
56:15DDX wherever you have a DDX that's where
56:18you're doing derivatives that's where
56:19you're doing calculus all right so if
56:20you show that it's going to let you know
56:22what you do next so in our case we'd
56:25have
56:26derivative of Ln no problem one over the
56:28inside times the derivative of the
56:31inside this lets you know where you're
56:32doing the calculus next this you're good
56:34right there's no DDX so you're fine this
56:37you need to keep going take a derivative
56:38and then we'll put it together and be
56:39done what is the derivative of 3x^ 2us 1
56:43what do we have there fantastic so we're
56:46going to
56:49have 1 over 3x^2 - 1 * 6 x where does
56:55the one go
56:56turns zero yeah sure the derivative of
56:58any constant always zero so we're good
57:00on that put it together make one
57:02fraction out of
57:09it I pray that you get the right answer
57:11here can I simplify
57:13anything can I start Crossing out sixes
57:16and threes it still happens sometimes I
57:18swear I still catch people this level
57:19yeah don't do that please don't do that
57:20unless you can factor out and simplify
57:22we're good uh do you all understand the
57:24first example yeah you okay with it not
57:27so bad right it's very much like chain
57:29wheel it's just you have a new function
57:31now it's not even a hard function either
57:33it's just you put one over the inside
57:35and and you're basically done let's see
57:37how much we can uh step this
57:50up okay that looks nasty enough I think
57:53we'll start there
58:00give me some ideas on what you could do
58:02with this uh true true or false let me
58:04ask you actually true or false can I
58:06move this cube root from the outside to
58:09the inside as a 1/3 power can I do that
58:12no no no no if it was 13 lnx absolutely
58:17but remember I told you about not making
58:18up your own mathematics don't make stuff
58:20up it's if it's not on one of those
58:21properties I gave you yesterday you
58:23can't do it so now I can't move move it
58:26here that's not what this is saying what
58:28this is saying is
58:29that this is supposed to be the
58:31derivative of lnx all to the 1/3 power
58:35do you guys see the difference there
58:37it's not here we can't we can't move it
58:39forward it's not an exponent of the
58:41inside of our logarithm of the argument
58:43of our logarithm it's that whole log X
58:45or lnx raised to the 1/3 power okay now
58:49you tell me what to do what do you think
58:51chain Ru chain rule I love the chain
58:53rule uh chain rule says to do you guys
58:56over here what what would you do
58:59here take the derivative outside take
59:04derivative outside do you guys see the
59:06similarity between the problem I just
59:08gave you and this problem right here or
59:11no I know there's an Ln over here but
59:13what we had here we had something to
59:16some power check it out if I cover this
59:18with my hand do I have something to some
59:20power and do the same thing what do you
59:22do with the 1/3 to the front sure
59:29I don't even care what's inside I know
59:31for a fact whatever's here if it's
59:33differentiable this 1/3 goes forward and
59:35something goes right here what goes
59:37right there 23 how do you get -2/3
59:40what's the operation
59:4613us on the inside what's going to go on
59:49the inside do we take the derivative of
59:52Ln now or do we take it later later okay
59:55so this we leave alone just like we left
59:58this
59:59alone we leave this alone but we're not
1:00:02done what's the next thing we're going
1:00:03to
1:00:05do perfect so we're going to show that
1:00:07right now DDX
1:00:15lnx so we understand well a cube root
1:00:18yeah it's still a 1/3 power but it's not
1:00:20a 1/3 power on the X3 power of the
1:00:23entire function here what that means
1:00:25that we we have a chain rule we have a
1:00:26general power rule we bring the one thir
1:00:28down no problem we taking a derivative
1:00:29we subtract one from it but then we
1:00:31don't forget that this stays the same on
1:00:34the inside of our function and we
1:00:35multiply by the derivative later of the
1:00:37inside of our function now this should
1:00:39be an easy one what's the derivative of
1:00:40Ln
1:00:41X fantastic so we got
1:00:461/3 Ln x^
1:00:49-23 * 1
1:00:54/x you know we can clean this up just a
1:00:57little bit of course this is a -2/3
1:00:59which means that you can take this whole
1:01:00thing and put it where yeah we can do
1:01:03that we can change it to a cube root of
1:01:05this to the second power if you want so
1:01:08this you might be done with the calculus
1:01:11but clean it up just a little bit don't
1:01:13leave it so nasty we could have
1:01:16one 3x no problem we can have a cube
1:01:20root of lnx to the 2
1:01:27that's how I'd like you guys to write it
1:01:28make it clean uh typically sometimes I
1:01:31let you do it with negative exponents
1:01:33most of the time I don't want negative
1:01:34exponents I want them nice and neat uh
1:01:36the only time that that's not the case
1:01:37if it's just a super huge uh derivative
1:01:41and it would just waste your time to do
1:01:42it so that's an appropriate answer show
1:01:44hands feel okay with that one fun right
1:01:47right lie to me right
1:01:51okay all right let's keep
1:01:54going so we know how to take derivatives
1:01:57of lnx we now know that the chain Rule
1:01:59still applies to
1:02:04that let's keep on working with
1:02:09it this is kind of good practice because
1:02:11it gets you back in the habit of doing
1:02:15derivatives and you're learning
1:02:16something new
1:02:30what do you think we're going to do
1:02:32first what do you think is this like a
1:02:35general power rule like we had last time
1:02:38is it a chain Rule still yes anytime you
1:02:40got a function instead of a function
1:02:42it's a chain rule pretty much anytime
1:02:43you do a derivative it's a chain rule
1:02:44sometimes it's easy because you just
1:02:45have X but it's always some sort of
1:02:47chain rule so we have Len of some
1:02:49function uh tell me people in the middle
1:02:51what's the first thing that I should
1:02:53write here up the L
1:02:56okay let's talk about it can we break up
1:02:59the Ln can we break up that natural laog
1:03:01oh remember that we can break up
1:03:04multiplication division and powers of
1:03:07the entire function inside the Ln can we
1:03:10break this one up can you break up
1:03:12addition so then we can't touch
1:03:15this can't touch
1:03:18this anyway we can't break this up
1:03:21because of that nasty plus there uh Ln
1:03:23of something plus something does not
1:03:25equal l n of something plus Ln of
1:03:27something it doesn't work if that had
1:03:29been multiplication of course we could
1:03:31but we can't do it here
1:03:32gcy okay so now we got to do the
1:03:35derivative like it is right now what
1:03:38would I do first one over the infin yeah
1:03:41that's fantastic this is a chain rule
1:03:42right the derivative of Ln says you put
1:03:45one over whatever this is I don't care
1:03:47what it is that's how the chain rule
1:03:49works for Ln so the first thing
1:03:51absolutely is one over just that nasty
1:03:55stuff
1:04:01do you guys understand that that is the
1:04:03derivative of Ln that's what that says
1:04:05Ln just says one over the inside that's
1:04:08all it means okay now to deal with the
1:04:11inside of it you use the chain rule so
1:04:13what are we going to do
1:04:17next the the inside
1:04:24perfect I'm going to change that right
1:04:26now so I have to do it again instead of
1:04:29the square root let's just make that the
1:04:301/2 power right
1:04:37now can you guys do that for me once
1:04:39youall do that take care of that
1:04:42derivative I'll write it on the board I
1:04:43want you guys to try it on your own
1:05:23I want to see if you got it did you get
1:05:24it did you guys notice that there's a
1:05:27double chain rule in here it's awkward
1:05:29right but that happens a lot in calculus
1:05:31we deal with one chain rule boom we got
1:05:33another one in there derivative of 2x is
1:05:36of course two we bring down the 1/2
1:05:39power bring down the 1/2 power leave the
1:05:41inside alone subtract one from our power
1:05:45and then multiply by the derivative of
1:05:47the inside um this sort of thing you can
1:05:49do without showing me the extra step I
1:05:50want to see the extra I want to see this
1:05:52stuff for Ln because that's what we're
1:05:54learning but those other the ones that
1:05:56you should know already feel free to
1:05:58just do them quickly okay uh I don't
1:06:00expect that we have any issues with this
1:06:02I hope do we well I hope I hope not do
1:06:05you do you okay if you do you let me
1:06:09know I can cover this a little more
1:06:10thoroughly uh do you guys see what I'm
1:06:12talking about though the chain rule at
1:06:13work we have bring down leave it inside
1:06:16uh subtract one and multiply by the
1:06:17derivative inside after this it's kind
1:06:20of your job to clean this up I'm not
1:06:22going to do any more work on this
1:06:24particular problem what I suggest this
1:06:26this gets really messy really fast when
1:06:28you have multiple chain rules especially
1:06:30with these negative exponents this right
1:06:33here of course you can move this to the
1:06:35denominator I'll show you one more
1:06:40step but the rest of it honestly is just
1:06:43some
1:06:44algebra you could do 3x^2 over 2 < TK x
1:06:49cubus one do you see how that works yeah
1:06:51we can bring this down that's a negative
1:06:53power that's a square root X - 1 this 2
1:06:56is on the denominator 3x2 goes numerator
1:06:59of course you have to show me the
1:07:00parentheses there that's important but
1:07:02in order to make this one fraction you'd
1:07:04have some work to do you'd have to get
1:07:06common denominator here and then put
1:07:09those two things together that's kind of
1:07:10up to you um I would say that this would
1:07:13be fine for for one of my answers in
1:07:15class uh in the back of the book like
1:07:17when you're checking your work or
1:07:18whatever it might not look like this
1:07:21most of the time they do the extra work
1:07:23uh they make the comma denominator they
1:07:24distribute they combine like terms they
1:07:26make one fraction out of it so if you
1:07:27want to get yourself really good at
1:07:29algebra try doing stuff like that you
1:07:32guys okay with this one so far so
1:07:33calculus is done the only thing you'd
1:07:34have to do now just a little bit of
1:07:36algebra y'all have any questions or can
1:07:38I move
1:07:39on move on WE you good yeah am I going
1:07:42too fast for you you okay all
1:07:47right tell you what we'll do about U
1:07:50three more of these just to get your
1:07:52heads wrapped around some of these other
1:07:53Concepts I want you to know uh we'll
1:07:55talk about a new technique that uh you
1:07:58probably haven't ever seen before we'll
1:08:00talk about implicit differentiation then
1:08:02we'll do log
1:08:14LS
1:08:23oops oh dear here we go
1:08:34first things first here let's look at
1:08:35our problem uh when you're doing
1:08:37derivatives or integrals it's really
1:08:39good to spend some time and diagnose
1:08:40what's going on sometimes these are
1:08:42going to be pretty easy but you still
1:08:43want to think about them what's the big
1:08:45idea on this derivative does Ln contain
1:08:48everything in this problem then it's not
1:08:50a good idea to start with Ln first it's
1:08:53a first good idea to start with say it
1:08:54again product yeah yeah a product
1:08:56contains all of this you guys see it we
1:08:58have X Cub anytime you have something in
1:09:00front of Ln works just like something in
1:09:01front of parentheses is being multiplied
1:09:03so there's a product right there do you
1:09:06you remember you remember remember the
1:09:08product rule yeah show me the product
1:09:10rule here go for it write it out
1:09:46product has a p in it plus has a p in it
1:09:49so product plus so product will you're
1:09:51going to have to add something together
1:10:04so I'm going to show my work that way
1:10:05you see where things coming from product
1:10:07work works like this you do the
1:10:09derivative of the first times the second
1:10:11plus the first times the deriva of the
1:10:12second some people do it backwards some
1:10:13people do the first times der of the
1:10:15second plus the second times der of the
1:10:17first or whatever I prefer this way just
1:10:20that's the way I grew up learning it
1:10:21derivative of the
1:10:23first times the
1:10:28second plus the
1:10:31first times the derivative of the
1:10:37second did you remember it if you didn't
1:10:41that's okay we're just starting here but
1:10:43relearn it right now because you're
1:10:45going to use it a lot all right again
1:10:48say what now say that again the
1:10:49derivative of the first time the second
1:10:51derivative of the first * the second
1:10:53plus the derivative of the second Plus
1:10:55time first sure yeah something like that
1:10:58something like that it's one of those
1:10:59things derivative of the first times the
1:11:01second plus the first times the
1:11:02derivative of the second derivative of
1:11:05the first one times the second one plus
1:11:10the first one leave it
1:11:11alone times a derivative of the second
1:11:14one so you take a derivative of one of
1:11:16them you leave one alone add take a
1:11:18derivative of the other one leave the
1:11:20other one alone oh the quotient is the
1:11:23quotient first happens when you have a
1:11:26right uh low D High minus high D low
1:11:29Square the bottom away we're done wait
1:11:31that's the quo rule that's the quo rule
1:11:33I remember it works for
1:11:36quotients product rule for products but
1:11:39product rule you're your derivative of
1:11:42the first times the second plus the the
1:11:44first times the Der of the second add
1:11:45them together that's the idea I don't
1:11:47really care the order in which you go U
1:11:49make sure you're taking the derivative
1:11:50of one of
1:11:52these factors multiply by the other one
1:11:55one add take a Dera of the other Factor
1:11:57multiply by the first one that's the
1:11:59idea now the derivatives themselves
1:12:02should be pretty straightforward uh you
1:12:04let me know X Cub what's the Dera of X
1:12:06Cub 3 sure so 3x2 you'll notice we leave
1:12:11Ln 5x alone we don't do anything with
1:12:14that then we have this add we have X
1:12:17cubed time how about the derivative of
1:12:21Ln 5x what do we have to use there
1:12:25chain rule so chain rule here says what
1:12:30perfect
1:12:33times sure I'm going to show it right
1:12:35now just so we get used to doing that
1:12:36times the derivative of
1:12:395x some people forget to do this if you
1:12:41forget to do this you can change your
1:12:43problem a lot you have a a one over five
1:12:46instead of 1x okay you you have
1:12:49something you have something different
1:12:50here
1:12:53so three x^2 Ln 5x no problem plus how
1:13:00much is
1:13:02that what happens to our fives that's
1:13:05what I mean about you have a 1 when
1:13:07you're not supposed
1:13:08to how about our
1:13:14X's simplify the things that you can so
1:13:17if we have x to the 3 here and we have
1:13:19an X down here we can simplify that
1:13:25sorry I know I I did a whole lot of
1:13:27simplification all at once uh but if you
1:13:29need this one more time the derivative
1:13:31of 5x is 5 simplify the fives we still
1:13:36have an X we have x to the 3r that makes
1:13:38x to the 2 show F be okay with that one
1:13:41what's one more thing you could do if
1:13:43you felt like doing some extra math here
1:13:44what could you do take out the X squ
1:13:47good you could Factor it that's right
1:13:48you could Factor this out and get 3 Ln
1:13:515x + 1 x^2 on the outside uh do I care
1:13:54not really if you get a different answer
1:13:56in the back of the book maybe that's
1:13:57what they do uh so just be aware of that
1:14:00can we move on do you guys have any
1:14:01questions at all are you getting your
1:14:03brains a little bit un cobwebbed with
1:14:05some of this stuff it's going to take
1:14:06some time it is uh but don't worry about
1:14:08it right now you will you'll get there
1:14:10okay
1:14:28derivative Ln absolute value cosine x
1:14:32one thing I want you to notice is that
1:14:34sometimes we have absolute value around
1:14:36our function and sometimes we don't um
1:14:39really when it comes down to it don't
1:14:40worry about it because when I gave you
1:14:42the definition the very first time you
1:14:44don't need to write this you already
1:14:45have in your notes when I gave you the
1:14:46definition the first time I really gave
1:14:48you the definition of Ln absolute value
1:14:50of x is 1
1:14:52/x so we don't really need to worry all
1:14:55that much about those absolute values
1:14:56it's just there to make sure that you
1:14:58it's telling you hey don't worry you
1:15:00don't have any negatives inside of your
1:15:02natural log function because we can't
1:15:04have those of course cosine has positive
1:15:06and negative correct so when they give
1:15:08this to you they're saying we're just
1:15:09focusing on positive cosine that's what
1:15:11we're talking about all right so don't
1:15:13worry too much about that but let's
1:15:15worry more about doing this derivative
1:15:19first thing Ln is the overarching
1:15:21function it contains everything here so
1:15:22let's take the derivative of the Ln you
1:15:24got y'all should be pretty good at this
1:15:26by now we've done a few examples what's
1:15:28the first thing I'm going to
1:15:29write one over cosine X is absolutely
1:15:31true yeah for sure 1 / cosine X you'll
1:15:35notice there's no more absolute value
1:15:37the derivative of absolute value X is 1x
1:15:40so that goes away don't no need to worry
1:15:42about that one are we
1:15:44done no this is what again sure so we're
1:15:48going to do the derivative
1:15:55we should be pros of this derivative of
1:15:56cosine X come on come on oh I heard some
1:15:59people say the nasty one and some people
1:16:01say the right one I know it's it's it's
1:16:03sign somehow right is it positive or
1:16:05negative negative very good so this is
1:16:08going to be 1 over oops that's not
1:16:14right 1 cosin x
1:16:17* sinx let's put that together we got
1:16:22sinx over cosine
1:16:26X say it again perfect that's tangent
1:16:30this this is tangent so this isga tan
1:16:36X cool that was interesting wasn't it so
1:16:39sometimes these logarithm functions of
1:16:41sign functions gives us other sorry
1:16:43logarithmic functions of U trigonometric
1:16:45functions gives us other triat functions
1:16:47it's kind of weird but a lot of those
1:16:48relationships show up with Ln it's
1:16:50really interesting can we do one more
1:17:00is this good practice for you are you
1:17:02getting something out of it I hope all
1:17:04right good I don't want to waste your
1:17:05time or my
1:17:16time should I stop there
1:17:23no don't worry I haven't gone
1:17:29crazy I know right it's
1:17:36crazy now I think you'll agree that
1:17:38looks pretty bad to start with right
1:17:41does to me at least go my goodness
1:17:43gracious that's not fun well I want you
1:17:45to think about what we can do and why we
1:17:47did the whole yesterday's lesson on uh
1:17:49learning how to expand and combine
1:17:52logarithms in every other example that
1:17:53we had before
1:17:56we can't expand
1:17:57it can't we could have but it really
1:18:00wouldn't have done anything here because
1:18:01that's pretty easy we couldn't really
1:18:03expand it for anything reasonable
1:18:05definitely can't expand that can we
1:18:07expand this one the more that you do the
1:18:10easier that your derivative becomes so
1:18:12expand these as much as you can then do
1:18:15derivative of each little piece man
1:18:17these become way easier uh I'm I'm going
1:18:20to kind of cheat here I'm going to do
1:18:21the whole expansion in my head right now
1:18:23hopefully you can follow me along we're
1:18:24going to talk about it though so stick
1:18:26with
1:18:27me notice the LNS over the entire thing
1:18:30you need that what do you do with this
1:18:33division how do you separate that so
1:18:36we're going to subtract off another Ln
1:18:37of this piece are you with me yeah okay
1:18:40what are we going to do about this
1:18:41product right here so we're going to
1:18:44subtract or we're going to add another
1:18:46Ln on there so we're going to have Ln of
1:18:48this plus Ln of this minus Ln of this
1:18:52okie dokie maybe I'll do two steps for
1:18:53you
1:19:02notice one thing um when you're breaking
1:19:04up your Ln or your logarithms you're not
1:19:08doing derivatives you're not doing any
1:19:10calculus at all so you still need that
1:19:12DDX we're not doing anything right now
1:19:13except using the properties of logarithm
1:19:15so please maintain this until you
1:19:18actually take the derivative once you
1:19:19take the derivative of course get rid of
1:19:21it so we got Ln of X2 no problem plus +
1:19:25Ln 2x^2 + 1 all to the 3
1:19:29powerus Ln of 5 - x^2 under root we'll
1:19:36make sure that's in the bracket because
1:19:37we want to take the d entire thing let's
1:19:39make sure I'm right am I right this plus
1:19:42this one because we had a product minus
1:19:44this one because we had a quotient okay
1:19:46uh should I stop here or should I keep
1:19:48going keep going the more you do the
1:19:50easier this becomes so let's keep on
1:19:52going what should I do
1:19:55square square fantastic what else going
1:19:59to square root up square root okay so
1:20:01that's a 1/2 power I'm going to move
1:20:03that to the front Cube Cube to the front
1:20:06let's do this so every one of these
1:20:08arguments has an exponent let's move to
1:20:10the front
1:20:36let's look and see what we did um you
1:20:39should know that when you take derives
1:20:41you can take derivative of each
1:20:42individual piece as you go is this
1:20:44derivative easy or hard easy that's a
1:20:46piece of cake this one easy pretty easy
1:20:49involves a little chain rule but not a
1:20:50big deal this Oney also easy we just
1:20:54made something that looks really nasty
1:20:55into three really easy pieces really
1:20:58easy pieces that's nice try that when
1:21:00you get to those heavy duty ones in a
1:21:02home you go oh my gosh what in the world
1:21:04see if you can split them up now don't
1:21:06split up things that you're not supposed
1:21:08to for instance can I split this up no
1:21:12it's got a plus this one no it's got a
1:21:14minus I can't touch those but we
1:21:16definitely want to be putting our expon
1:21:18for it I definitely want to be
1:21:19separating products and quotients still
1:21:22okay so far okay question
1:21:28you I don't think so when you bring down
1:21:32that okay remember that right now we
1:21:34still have this DDX out front it's
1:21:35telling us we're not doing derivatives
1:21:37yet we're not touching calculus all this
1:21:40was was logarithm properties so all this
1:21:43was was hey remember how we brought the
1:21:44three down we're not subtracting it
1:21:46because that's where exponents go
1:21:48remember how that's a 1/2 power we're
1:21:50bringing down the 1/2 power we're not
1:21:51subtracting CU we're not doing a
1:21:52derivative right now now is the time
1:21:54where we take a d question I think what
1:21:57you subraction for the to it and it was
1:22:02exponent so would it be an
1:22:04addition like at the
1:22:07bottom no it doesn't matter um like when
1:22:10we did something like this Ln X Y we do
1:22:14Ln xus Ln y this takes care of the
1:22:18negative exponent for
1:22:19us do you see why you could actually
1:22:22write this Ln x * y y1 correct you can
1:22:26separate this by Ln X Ln y1 that's an
1:22:30exponent that goes down in front lnx
1:22:33minus Ln y That's why that works okay
1:22:35that's actually a little proof for you
1:22:37didn't want to have to show that but
1:22:39there you go does that make sense
1:22:45okay now we you to do derivatives what I
1:22:47want you to do is practice this do the
1:22:49derivative of this piece this piece and
1:22:52this piece it's nothing fancy it's just
1:22:54little chain rolles so show that to me
1:22:56and I'm going to do it over here for you
1:22:57okay
1:23:30do yall get it already yeah felt good
1:23:33all
1:23:35right you I got space
1:23:41over oh let me think so derivative means
1:23:45that because I can split up derivatives
1:23:47by addition I can do that for for
1:23:49derivatives uh I get to do derivative
1:23:52this and then this and then that so I
1:23:54know I've got 2 * the derivative
1:23:57remember with with constants you don't
1:23:59need a product rule uh if you did the
1:24:01derivative of two would be zero so that
1:24:03would kind of eliminate half of your
1:24:04product rule anyway you don't need it
1:24:06for constants only functions of x times
1:24:07functions of X so no product rules here
1:24:09on any of this 2 * 1X no problem 3 *
1:24:14again that's a constant we have 1 2x^2 +
1:24:19one one over the inside times the
1:24:23derivative
1:24:28of the inside three stays there
1:24:31derivative of Ln of this nasty junk we
1:24:33put one over the inside time derivative
1:24:36of the inside
1:24:37minus again we leave the2
1:24:40there derivative Ln 5 - x^2 well we do
1:24:44one over the
1:24:47inside time derivative of the inside
1:24:58let's see what we
1:25:03got I know I can put that together
1:25:05that's 2x I know that this what's the
1:25:07Der of that one so I got
1:25:104X * 3 that's going to give me 12x over
1:25:15this stuff none of it simplifies so
1:25:16that's going to be part of my answer
1:25:27you guys two for two did you get both of
1:25:29them Dam how much is this derivative
1:25:32what's that give
1:25:33you so this is let's write that this
1:25:36would be 4X that's how we got the 12x
1:25:40this you said this is
1:25:42what okay so if we put this
1:25:46together hopefully I'm not going too
1:25:48fast for you
1:25:50-2 pos2 minus what's the sign going to
1:25:55be for this fraction posi yeah sure
1:25:57minus and negative gives us a plus 2
1:26:00over 2 those simplify out so we're not
1:26:03going to have a a two in here we're
1:26:05going to have one over this that's it
1:26:09X say what now oh yeah X X's aren't
1:26:12really that important are they I
1:26:14mean yeah okay you're right I forgot so
1:26:17X over 5 - x^2
1:26:24did I miss anything else let's let's
1:26:26double check 4X * 3 that gives us our
1:26:2812x this stuff doesn't change twos are
1:26:31gone minus with the negative gets to
1:26:34plus X over 5 - x^2 we got that that's
1:26:37as good as I want how many people got
1:26:39that one that's good if you did that you
1:26:40know what you're doing okay um if you
1:26:42didn't if you struggled a little bit
1:26:44this is one of those times when before
1:26:46you start your homework you should go
1:26:47back through your notes do the problems
1:26:50that we've done in class like don't just
1:26:52follow them actually do them like
1:26:54different sheet of paper put your notes
1:26:55aside see if you can do them do them
1:26:57until you can get it right and then
1:26:58start your homework I promise it'll be a
1:27:00whole lot easier if you can do that okay
1:27:01if you jump right into your homework you
1:27:02go I'm a little overwhelmed because you
1:27:04never really learned the material you
1:27:06need to learn the stuff before you try
1:27:08the homework okay you with me the
1:27:10homework's there to kind of beat it into
1:27:12your head so you get the repetitive
1:27:14nature of
1:27:16this okay are you ready for something a
1:27:18little bit
1:27:19new this thing this is cool I promise
1:27:22you're going to like this this is this
1:27:24is nice you you really will I do at
1:27:26least I think you like you're going to
1:27:27like it um what this is called this next
1:27:30little piece we're going to talk about
1:27:31is called logarithmic differentiation so
1:27:34when they ask you for that and you go
1:27:36what in the world is logarithmic
1:27:38differentiation that sounds horrible
1:27:40it's really not horrible it's actually
1:27:41not fun uh this is what they're talking
1:27:44about and it stems from this idea let's
1:27:46suppose that we have some sort of a a
1:27:50function like this one
1:28:03question is this still 6.1 sure is I'll
1:28:05let you know when we
1:28:07change we got a function like this one
1:28:10that well we could probably take the
1:28:12derivative of that couldn't
1:28:14we just like it is I think I could you
1:28:19guys had it earlier what you could do
1:28:20what what what could you do here quo
1:28:22rule would have to be done then then
1:28:24you'd have a chain Rule and you'd have a
1:28:26chain rule but you can do it uh it's not
1:28:28going to look very pretty but you could
1:28:30you could definitely do it but there's
1:28:32another option if you have some of these
1:28:34derivatives you go man that's it's kind
1:28:36of nasty and they get way this is kind
1:28:37of an easier example to do this with but
1:28:39then get really nasty okay especially if
1:28:41I had like another function of X up here
1:28:43you'd have a quotient rule then a then a
1:28:45product rule inside the quotient rule
1:28:47then some chain rules it' get nasty
1:28:49really really fast sometimes we can get
1:28:51around that and here's how you do it the
1:28:54way that you do is called logarithm
1:28:55differentiation what you need to know is
1:28:57that you can do anything you want as
1:29:00long as you do it to both sides of an
1:29:01equation practically anything you want
1:29:03well one of those things that we could
1:29:05do looks a little weird but one of those
1:29:07things we can do we can actually take an
1:29:11Ln of both sides of our equation can
1:29:15someone out there explain to me why this
1:29:17would actually help us
1:29:20here
1:29:22aha if I taking Lon on both sides this
1:29:25doesn't help us very much but this does
1:29:28because I can separate it I can expand
1:29:30it all out are you with me yeah okay so
1:29:34what we're going to do before you start
1:29:36showing me derivative before you start
1:29:38showing DDX or doing anything like that
1:29:39you're going to expand this thing so no
1:29:41DDX is right now we expand it first so
1:29:45what that means
1:29:46is Ln y equals well let's see what we
1:29:50can do we got a quotient here what's a
1:29:51quotient change to everybody real quick
1:29:54subtraction very good do we have any
1:29:55products here no but we have some
1:29:57exponents we have a four and we have
1:30:00a cool so I'm going to do all that right
1:30:03now so we got
1:30:05Ln 4 * Ln of x
1:30:08-1 -3 Ln 2x
1:30:14-1 can you guys double check me make
1:30:16sure I did that right okay
1:30:18cool is that as expanded as we can
1:30:22get yes
1:30:24can we break this
1:30:26up please please goodness say say no can
1:30:29you break that up how about this one no
1:30:31we can't we can't separate subtraction
1:30:34so this four came forward no problem
1:30:36this we have subtraction 1/3 came
1:30:38forward no problem now we're ready to
1:30:40take a derivative do you remember doing
1:30:42implicit differentiation y Prime yeah
1:30:46we're going to have y primes everywhere
1:30:47you touch a function of Y you get that y
1:30:49Prime I like the dydx because that's how
1:30:51I grew up with the liven this notation
1:30:54but here's the idea check this out with
1:30:56me we're going to do a
1:30:59derivative of both sides so
1:31:03here and
1:31:08here you'll notice that our derivative
1:31:11is with respect to
1:31:12X which means that we treat y as a
1:31:16function of
1:31:18X what that means is that we do our
1:31:20derivative just like normal so the
1:31:22derivative of Ln
1:31:24Y what would the Der of Ln y
1:31:27be very good but check this out because
1:31:30this is a function of X we have to show
1:31:33the chain rule it is a chain rule here
1:31:35so what this looks like is 1 over y for
1:31:37sure but check this out why you get the
1:31:39Y Prime why and I don't really really
1:31:41like the Y Prime because notation wise
1:31:43it kind of fails you here but notice you
1:31:46would do a
1:31:48derivative of the inside does that make
1:31:51sense you'd have hey one over y this is
1:31:54the the Der of Ln times the derivative
1:31:57of y look at this this is dy DX that's
1:32:00what that is that gives you your
1:32:01derivative so I'm going to go down here
1:32:04and give you Dy
1:32:08DX that's this side that that's all you
1:32:10got to do so we have how do you take a
1:32:13Dera l and y no big deal it's one over y
1:32:16just like always but then you do a chain
1:32:17rule derivative of y hey that's it
1:32:20that's your derivative this is what
1:32:21we're going to solve for in just a
1:32:22second now let's do the derivative of
1:32:24this side this should be pretty easy I
1:32:26want you guys to do it go for it
1:33:12we'll do one more and we'll call it good
1:33:14uh for today's
1:33:18lesson oh goodness uh oh did you were
1:33:20you able to do this derivative yeah
1:33:23should get
1:33:244 -1 did you get 4 -1 we do the chain
1:33:28rule but the derivative of x -1 is
1:33:30simply one so this is 4X -1 minus well
1:33:36we got a 1/3 here we've got a 1 over 2x
1:33:42- 1 because we're taking a derivative Ln
1:33:44but now we do have a chain rule don't
1:33:46forget about the chain rule my goodness
1:33:48derivative of the inside gives you what
1:33:51how many people made it that far that's
1:33:53cool that's great it's fantastic let's
1:33:54clean this up a little bit a couple
1:33:56things need to happen firstly we have a
1:33:584X -1 we've got a 2 over three don't
1:34:02forget parenthesis here we get a 2x-
1:34:06one can you distribute that just make it
1:34:09you can you can distribute it U
1:34:11generally people don't typically
1:34:13distribute denominators because if you
1:34:15have to create a common uh denominator
1:34:18it's easier to do it when it's Factor
1:34:20okay CU here you you distribute but then
1:34:22to find a common denom you have to
1:34:23Factory so typically people don't do
1:34:25that it's not a big deal if you do or
1:34:26not okay now the last thing if you'll
1:34:30notice this we haven't solve for our
1:34:32derivative yet we have 1 y * dydx what
1:34:36you need to do when you're doing this
1:34:37process solve it for dydx this stands
1:34:39for your derivative so if you get this
1:34:41by itself you're done tell me how can I
1:34:44get this piece by
1:34:47itself
1:34:50fantastic so if I multiply by y and
1:34:53multiply this whole thing by
1:34:59y well these are gone that's cool we get
1:35:04dydx equals I'm going to leave all this
1:35:07stuff the same 4X -1 2 over
1:35:113 2x -
1:35:151 time
1:35:18y okay I want to show of hands real
1:35:20quick if you're okay with what we've
1:35:22done so far in the problem
1:35:24good that's fantastic now question for
1:35:26you is it appropriate if I can change
1:35:29the Y should I leave the y the way it is
1:35:32or would you rather see function of
1:35:34X right now this is implicit if I can
1:35:37solve for an explicit derivative that's
1:35:39a little bit bit better look back at our
1:35:42problem do you know how much Y is equal
1:35:44to yes do you you should how much is y
1:35:54that that bunch of crap hey look at that
1:35:56we got it so all you got to do I know
1:35:59it's fun told you it was fun day sure we
1:36:02took L on both sides that's so that we
1:36:04can split all this junk up okay we split
1:36:07all the junk up we took a derivative
1:36:09after that notice here's our DDX this is
1:36:10where you do calculus calculus this side
1:36:12is implicit calculus this side is pretty
1:36:15easy put it together as much as you can
1:36:17solve for your derivative whatever that
1:36:19is if you used y Prime no problem solve
1:36:20for y Prime solve for dydx here by multi
1:36:23applying y on both sides now just make a
1:36:26substitution this should be explicitly
1:36:28defined somewhere in fact it's right
1:36:30here so this
1:36:33piece goes here if we do that then we're
1:36:41done um I really don't care how nice you
1:36:43make this you're going to find out that
1:36:45your derivatives can look quite
1:36:46different uh from the back of the book
1:36:49um right now I'm not trying to get you
1:36:51to be super excellent at algebra I'm
1:36:53putting this stuff together you really
1:36:54should know that so you should know how
1:36:55to find common denominator and put that
1:36:58together okay I'm trying to get you to
1:36:59be good at the calculus part of it um
1:37:01and that's that's really our our goal
1:37:03here so I'm going to leave it with these
1:37:05two fractions you need to know that you
1:37:07can manipulate it a little bit so we'd
1:37:11have 4x -
1:37:141 - 2 over
1:37:173 2x -
1:37:221 big old bracket and then we multiply
1:37:26by however much the Y is equal to since
1:37:29the Y is this stuff that's what we put
1:37:31on the back
1:37:47end my test reasons will you mark us
1:37:50down for not doing the Y what it equals
1:37:52two or like but y absolutely I do want
1:37:56the Y Prime is is perfectly fine but I
1:37:58do need you to notice that this is a
1:38:00function of X this should be substituted
1:38:02back for I don't want you to leave it
1:38:03why I want you because this is what
1:38:05logarithmic differentiation is it's kind
1:38:06of going around the derivative but still
1:38:08getting a proper derivative in terms of
1:38:10X okie dokie are you guys okay with this
1:38:13idea question so if we leave it like
1:38:15that and don't do the algebra we're fine
1:38:17this is fine for me I need to tell you
1:38:19that in the back of the book uh
1:38:20typically what they'll do is they will
1:38:22find a common genom they'll put this
1:38:24together it's going to look a little bit
1:38:25different because they want one fraction
1:38:27if possible do you see what I'm talking
1:38:28about if they get one common denominator
1:38:31put this together then you could make
1:38:33one fraction out of it I'm not going to
1:38:34spend the time to do that because we
1:38:36could spend all day doing things like
1:38:38that so F be okay with it okay now the
1:38:41last thing we're going to talk about we
1:38:42got about two minutes here I want to
1:38:44show you something like this I don't
1:38:46know how much we're going to get through
1:38:48um I'll probably just go pretty pretty
1:38:51quick at it but I want you to think
1:38:52through this uh firstly can you think
1:38:54about this as an implicit problem do you
1:38:56guys see that we have x's and y's all
1:38:57tied together and we can't really get
1:38:59them
1:39:00separated okay there is one thing that
1:39:02we can do to make a problem easier what
1:39:03is
1:39:06it you have it it's in your head I can
1:39:08see
1:39:09it the Ln separate yeah the Ln
1:39:12absolutely let's separate that thing
1:39:14that makes our problem easier do you
1:39:16guys see it so we would do Ln 2x - Ln y
1:39:22- sin y + x^2 = 0 that's as much that's
1:39:28as easy as we can get it okay we can't
1:39:29do anything really substantial to make
1:39:31this problem any easier we could
1:39:33separate these but it wouldn't make a
1:39:35difference because that's a constant who
1:39:36cares derivative that thing going be
1:39:38pretty py so we would take a derivative
1:39:42of both
1:39:45sides now would it be easier to separate
1:39:50the x's and y's on their respective
1:39:53sides you can it doesn't really matter
1:39:55you're going to do it at one point or
1:39:56another okay so here I'm going to be
1:39:58kind of quick about this I I really
1:39:59don't want to keep you over so uh
1:40:01derivative of Ln 2x can you tell me what
1:40:03that is 2 2x sure it's 2 2x it's 1 2x *
1:40:092 because we have the chain rule here we
1:40:11have -1 over y * y Prime if you like the
1:40:14prime notation that's y Prime minus
1:40:17derivative of sign everybody der of sign
1:40:19real quick so we would have cosine y but
1:40:22check this out please please because you
1:40:24took a der of Y you're going to have a y
1:40:27prime plus this one der of x^2 is
1:40:32perfect D of zero cool 2x cosine y y
1:40:36Prime 1 y y Prime 2's are gone we get
1:40:401/x all you have to do now get all your
1:40:42x's on one side Factor your y Prime just
1:40:45like implicit differentiation does and
1:40:47you're done
1:40:48so let's see
1:41:00if we do that we're going to have a 1 y
1:41:02* y Prime cosine y * y Prime on the
1:41:07right hand side we have a -2X -
1:41:111X how' you get
1:41:14um what I do you put 1X Plus what are
1:41:18you talking
1:41:19about just
1:41:21checking as soon as you have your y
1:41:24Prime terms on one side and everything
1:41:25else on the other side this is when you
1:41:27want to factor out your y
1:41:36Prime and then just a
1:41:54so break I guess the long story short
1:41:57break down the LNS as much as you can
1:41:59break them down then start taking
1:42:01derivatives whether you need logarithmic
1:42:02differentiation okay whether you need
1:42:04implicit differentiation okay if it's
1:42:06implicit you're not going to get many of
1:42:07these but if it's implicit make sure you
1:42:10take and do a y prime or dydx everywhere
1:42:13you take a derivative of y get all those
1:42:16terms to one side everything else on the
1:42:18other side Factor y Prime divide
1:42:20typically and you're done um this this
1:42:23answer by the way it can look different
1:42:25you need to know something about this
1:42:26you can multiply and get a common
1:42:28denominator you multiply get a common
1:42:30denominator do a complex fraction on
1:42:32that and it's going to look a little
1:42:33different what I want to know is is
1:42:35everyone okay getting down to at least
1:42:37this far so you are yes is it making
1:42:39sense to you yes okay so we
1:42:43don't all right so now it's time to keep
1:42:45going with our our natural logarithms
1:42:47we've learned all about the derivatives
1:42:49we did a lot of examples from that now
1:42:51we're going to start talking about
1:42:52integrals so if you remember the first
1:42:54thing I actually proved other than just
1:42:56the properties of logarithms was the
1:42:58integral and derivative of that so what
1:43:00we should have down I think you guys
1:43:02already have this but we'll talk a lot
1:43:04about it
1:43:11today our goal was to find out what in
1:43:15the world happened when I
1:43:18take an integral of something like 1 / U
1:43:22we tried that right right we did it at
1:43:23first and like man this didn't work like
1:43:25the normal way we we were to do this we
1:43:27can't make this U to the1 and then add
1:43:30one to it because we get U to the 0o we
1:43:32get one over Z that's a big problem so
1:43:34we said well well how else how are we
1:43:36going to do that so we did the uh the
1:43:38whole Ln thing we said well this really
1:43:41this really
1:43:42is Ln of the absolute value of U plus C
1:43:46do you remember the absolute value that
1:43:48we had okay so that's the integral that
1:43:50we're we're talking about integral of
1:43:51one over U gives you Ln absolute value
1:43:54of U plus C that's what we're working
1:43:56with today what I want to do is just
1:43:59practice a few very basic examples with
1:44:01a little bit of substitution for right
1:44:03now and we'll gradually gradually work
1:44:05our way up to some trig functions um and
1:44:08see how that works for us you guys ready
1:44:10yeah
1:44:13so first
1:44:20thing let's consider in integral of 1 5x
1:44:24- 2 DX now if you remember back to your
1:44:28integrals your idea was and we haven't
1:44:31really learned any integration
1:44:32techniques yet you're still in the same
1:44:35boat the same the boat being you either
1:44:38have to have something that's directly
1:44:40on your table remember the integration
1:44:42table you guys have I know some of you
1:44:43had the little sheet out right now uh
1:44:46either if it's something directly on
1:44:47your table like X 3r or just secant
1:44:51squar or something like like that or you
1:44:54have to substitute to make it fit your
1:44:56table well here's one more thing that we
1:44:58have added to our table so if we can
1:45:01make it look like this then we can do
1:45:03the integral does this look exactly like
1:45:05this no it doesn't so if it doesn't look
1:45:08like that we've got to make it look like
1:45:09that what's a good way to make it look
1:45:11like that subti substitution is
1:45:13fantastic remember how substitutions
1:45:16work substitutions work when you look at
1:45:18a part that you don't like in your
1:45:20problem like this I don't like that part
1:45:23and the derivative of this part has to
1:45:25be somewhere in your function
1:45:27disregarding the constants so what's the
1:45:30derivative of this five but there's no
1:45:32X's right so that's cool we can pull
1:45:34fives out in front that's the idea we're
1:45:36going to make a substitution to make
1:45:38this guy look like this guy so our
1:45:40substitution here has to
1:45:47be U = 5x - 2 let's see well oh you know
1:45:51what I want you to practice that too
1:45:53what do we do when we do a substitution
1:45:54what's the next
1:45:55step yeah okay cuz we got to we have to
1:45:58not only substitute for this but we've
1:46:00got to change this as well this has to
1:46:02match up with whatever variable you are
1:46:04substituting for so yeah we need a du
1:46:08equals mhm 5 DX um I learned it this way
1:46:12you guys can do your substitution
1:46:14however you want I learned to solve for
1:46:16DX so you'll always see me doing du over
1:46:185al
1:46:20DX some people have done it other way
1:46:22it's fine I don't care what way you do
1:46:24it okay uh so we'll do a
1:46:26substitution we still have a one uh what
1:46:29does the 5x - 2 become good and instead
1:46:33of DX I
1:46:34put
1:46:39perfect what do you typically do with
1:46:42that five move it out to the yeah that
1:46:44way it doesn't affect any of your makes
1:46:47it more difficult or anything makes it a
1:46:48little easier so 1/ integral of 1 U du
1:46:52we want to make sure this matches this
1:46:55your variables have to
1:46:57match did we make it look good now does
1:47:00it fit part of our table y that's
1:47:02fantastic that's the whole idea here so
1:47:04uh don't trick yourself and thinking
1:47:05these things are ridiculously hard right
1:47:06now pretty much all you have all you
1:47:08know is it either fits your table or you
1:47:11substitute to make it fit your table
1:47:13that's about it so we did a little
1:47:17substitution I know the 1 stays there
1:47:20that doesn't affect anything everybody
1:47:21integral of one to U du
1:47:24is
1:47:26Ln good absolute value is important plus
1:47:29C sure you put that now or you can put
1:47:31at the very end it really doesn't matter
1:47:32there is one thing that is important
1:47:34though what variable did we start with
1:47:36an x what variable do we have now that's
1:47:39a problem so what are you going to do
1:47:41put X back in yeah so if we have this
1:47:43substitution U = 5x - 2 that U becomes
1:47:485x -
1:47:492 so our integral is 1
1:47:55Ln 5x - 2 in absolute
1:47:58value plus C okay that's our first
1:48:01example it's a really simple one but I
1:48:02want to make sure that you guys are okay
1:48:03with it show fans if you are you feel
1:48:05okay with that one perfect fantastic
1:48:08let's step it up just a little bit I
1:48:10want you to know that we're not always
1:48:13substituting just to make an Ln
1:48:15sometimes we're going to have a sorry I
1:48:17one over you to get a integral that
1:48:18equals Ln sometimes you'll have Ln in
1:48:20our integral so let's practice one of
1:48:22those to
1:48:35all right looks a little scary but let's
1:48:38look at
1:48:40it the first thing you should do
1:48:42whenever you do an integral is to see if
1:48:43it fits your table is this anywhere in
1:48:45an integration table that you know of no
1:48:48it looks pretty nasty actually there's a
1:48:50square root there's an lnx it's got a
1:48:51fraction that's all the bad stuff at
1:48:53once right that's no good but then we
1:48:56think if it doesn't fit the integration
1:48:57table we should look to make uh sub
1:49:01that's right let's make a substitution
1:49:02here the appropriate substitution is one
1:49:05that's typically inside another function
1:49:09it's usually like a denominator or it's
1:49:10inside parenthesis or it's inside a root
1:49:13it's usually inside something and the
1:49:14derivative has to be there disregarding
1:49:16constants so let's look at the
1:49:18appropriate substitution is the
1:49:19appropriate substitution 3x no no the
1:49:22derivative of that is three I don't
1:49:24that's not going to help us with our Ln
1:49:26is the appropriate substitution the
1:49:27square root of
1:49:29lnx with the square root yeah it's
1:49:32typically not the whole function guys
1:49:33it's typically the inside of a function
1:49:35you you take the ugly part out now
1:49:37here's what I mean by that
1:49:38look if you had a u in here if you had a
1:49:42u could you take the integral of U to
1:49:45the 1/2 power yeah so don't worry about
1:49:47that square root also the derivative of
1:49:50lnx I'll think back to yesterday the Der
1:49:52of Ln X
1:49:55is here's a one do you see it here's a
1:49:571x that's something that we're looking
1:49:59at so you want the derivative to be in
1:50:01there and typically they use the inside
1:50:03of some other function so let's try
1:50:06it = lnx and you already told me that
1:50:09when we take our derivative we're going
1:50:10to get du = 1
1:50:13/x
1:50:15DX again I always solve for DX you don't
1:50:18have to but you can If I multiply both
1:50:20sides by X I get X
1:50:24dual DX do you guys see what I'm talking
1:50:26about about your derivative has to be
1:50:28there because right now when we
1:50:30subtitute this in for DX our X's are
1:50:32going to go by by and what you have to
1:50:34have is all one type of variable and
1:50:36that's got to match up your du or DX or
1:50:38whatever you have you guys ready to do
1:50:40the rest so let's see uh you tell me
1:50:42what I'm going to have do I still have a
1:50:44square root yes sare root of you inside
1:50:46that square root is
1:50:49M denominator I got a 3X still I haven't
1:50:52changed changed that yet but I have an
1:50:55X
1:50:57du yeah that's right the x is due so X's
1:51:01are gone that's fantastic we had to have
1:51:03that happen if that doesn't happen
1:51:06you've made the wrong substitution okay
1:51:07you have to get rid of all the other
1:51:09variable or be able to substitute for it
1:51:11in a different way you have to have
1:51:13these things match up and have no other
1:51:14extraneous variables going on tell me
1:51:16one other thing two other thing that
1:51:18things that we're going to do on this
1:51:20problem 1/3 goes out what
1:51:25else perfect square root to 12 mhm so
1:51:291/3 no problem we've already got rid of
1:51:31our X's that was a big deal We've Got U
1:51:34to the 12
1:51:35power du be careful when you're doing
1:51:38your integrals remember that we're
1:51:39always adding one we're not subtracting
1:51:41that's a derivative so when we do our
1:51:43integral here we got 1/3 no problem
1:51:45we've got U to
1:51:47[Music]
1:51:48the 2 23 Goodtimes 2/3 where we getting
1:51:52the 2/3 from I'm going to show it here
1:51:54this would be over 3es but then what you
1:51:57know is you got a complex fraction the
1:51:59two just moves to the numerator so
1:52:02really we've got 1/3 *
1:52:0523 U to the
1:52:083es nothing simplifies here we've
1:52:13got hopefully I did it right 2 n u to
1:52:183 plus C yes and we're done right you
1:52:23got to put the we're not done we're
1:52:26close to done we're almost done the last
1:52:28thing we got to do is make sure that if
1:52:29we've done a substitution you got to
1:52:31substitute back for it so since we did
1:52:33all the way back over up here we're
1:52:35going to put this back for this that
1:52:37we'll be done by the way is my math all
1:52:39right yes okay good so we've got 2
1:52:449ths yeah lnx we'll just move that three
1:52:48halves up like that around the whole
1:52:50thing uh one one quick question can can
1:52:52I move this three Hales to the front yes
1:52:55or
1:52:55no yes or no what do you
1:52:58think
1:53:03can okay so question is this the same as
1:53:09this no no okay this one you can this
1:53:13one you can't this is the whole function
1:53:15raised to the power does that make sense
1:53:17to you yeah okay so be careful on that
1:53:19again don't invent your own mathematics
1:53:21please
1:53:23plus C that's done show be okay with
1:53:25that one boom
1:53:28fantastic that's really I'm only going
1:53:30to give you two of these ones they're
1:53:31pretty basic you're typically going to
1:53:33use a substitution you're either going
1:53:34to have something like one over U uh to
1:53:37substitute or sorry after substitution
1:53:39or you can substitute lnx see if you can
1:53:41work around with that of course I can't
1:53:43give you every type of example you're
1:53:44going to have but these two cover a lot
1:53:46of bases here so look for substitutions
1:53:49try to make it fit something that's in
1:53:50your integration table now the next one
1:53:53next one the now now gets fun this is
1:53:55the good stuff I know you've been
1:53:56waiting for it I know you've been
1:53:57waiting for the trigonometry right right
1:53:59am I right yeah I
1:54:03know well so
1:54:09far here's what here's a couple of them
1:54:12that you know we
1:54:16know this
1:54:18one and this one
1:54:23and this
1:54:27one and we
1:54:31know I going just do a few just these
1:54:34ones for right now you know this one
1:54:36let's practice see what you guys know
1:54:39um let's start with this one what's
1:54:46that how about that
1:54:48one good oh you guys are good at
1:54:51this some of you in the math lab took my
1:54:53advice and actually learned these when I
1:54:55told you to that's fantastic how
1:54:57about good
1:55:04D
1:55:06okay thanks one of
1:55:12us there's a couple other ones uh what's
1:55:16another give me another one I think you
1:55:17got cosecant cotangent yeah that's the
1:55:21one and then
1:55:29squ what's this
1:55:31one that's
1:55:35right and then the last one should be uh
1:55:38cosecant squ right
1:55:43yeah let's see if secant s = Tan x cose
1:55:49s should equal Co something Coan
1:55:52negative negative that's
1:55:57right that right now is your integration
1:55:59table
1:56:00for trigonometric functions I believe
1:56:03that's about all that's on there right
1:56:04we're going to create some more right
1:56:06now so this is what you're kind of
1:56:08expected to know uh coming in this class
1:56:11everyone from your verbal responses knew
1:56:14these ones yay uh but these ones are a
1:56:17little bit harder to memorize because we
1:56:18don't use them as often we're going to
1:56:20use these okay we're going to use them
1:56:22so get used to them uh write this down
1:56:24have that in your your memory banks you
1:56:27okay okay now what we don't see up here
1:56:31is stuff like this one we don't see the
1:56:37integral of tan
1:56:40X if you've wondered why in the world
1:56:42don't we ever see the integral of tan X
1:56:44why couldn't we deal with that well
1:56:46we're going to find out right now why
1:56:47you couldn't deal with that in calculus
1:56:49one so uh give me some ideas on how we
1:56:51might go go about now that we know some
1:56:53of the stuff that we do know how we we
1:56:54might go about solving this particular
1:56:56intergal what would you want to do
1:56:58first good we use an identity uh you'll
1:57:01find out that with trig you use a lot of
1:57:04identities I wasn't lying to some of you
1:57:06who came to me earlier and said what do
1:57:07I need to know learn your identities
1:57:09learn your identities because we're
1:57:10going to use a lot of them so yeah let's
1:57:12bring it up tangent is sinx over cosine
1:57:16X go through the steps I'm talking about
1:57:19here now we're going to look at this
1:57:20does this fit doeses this fit over here
1:57:23exactly no does it everybody does it fit
1:57:26over here exactly no no it doesn't look
1:57:28exactly the same if it doesn't look
1:57:30exactly the same then you're going to
1:57:32use substition yeah you're going to make
1:57:34a substitution to make it look the same
1:57:37okay
1:57:38well let's do a substitution here
1:57:41now it's true that the derivative of
1:57:45sign is here but if I solved for it it
1:57:48wouldn't cross it it wouldn't simplify
1:57:49that out of the problem so you being
1:57:52sign's probably not a good choice We
1:57:54want to pick the worst part of this and
1:57:56call it u in our case right here we're
1:57:58going to pick U equals the
1:58:05coine cosine
1:58:08X do you guys see what I'm talking about
1:58:10about the worst part of this being uh
1:58:12being cosine is on the denominator also
1:58:14it's a great part for us because if I do
1:58:16the derivative the derivative of cosine
1:58:19is and besides the negative the sign
1:58:22parts right there that's something we're
1:58:23going to be able to simplify out so
1:58:25that's a good choice so
1:58:27dual
1:58:32s x
1:58:35DX you know me I do du over
1:58:40sinx =
1:58:42DX let's see what this
1:58:45does uh do we still have a
1:58:48sign well it depends on how you do the
1:58:50integral for me yeah I still do because
1:58:51I I like to do
1:58:53this do we still have a
1:58:55cosine what's our
1:58:58cosine do we still have a
1:59:00DX no du for me or sorry DX for me is Du
1:59:05over
1:59:06negative sinx our
1:59:09signs are
1:59:13gone tell me a couple things that we're
1:59:15going to do that so this is negative
1:59:19integral 1 / U
1:59:22man that made it a lot nicer can we do
1:59:24the integral no you can we can now
1:59:27because we learned about what the
1:59:30um we learned about that thing what's
1:59:32that thing we learn about Ln that's
1:59:35right we learned about natural law so
1:59:37notice we do have the
1:59:39negative Ln
1:59:43of you're forgetting something absolute
1:59:46value absolute value is absolutely
1:59:50important okay that's that's cool we're
1:59:52almost done let's go one more step I
1:59:54think someone else said that already you
1:59:57can't just leave it in terms of you what
1:59:58are we going to put in place of our
2:00:01U so this equals I'm going to go right
2:00:04over here so it looks better this equals
2:00:07negative
2:00:09Ln absolute
2:00:13value cosine X plus C show be okay with
2:00:16that one isn't that interesting did I I
2:00:19think I told you I think I prefaced this
2:00:20last time and said it's weird but a lot
2:00:23of trig functions get related to Ln
2:00:26somehow don't you it's interesting isn't
2:00:28it that when you take the integral of
2:00:30tangent you somehow get Ln of absolute
2:00:32value negative Ln absolute value of
2:00:34cosine you guys okay with that one it's
2:00:37pretty straightforward right now there's
2:00:40one more thing that we're going to do
2:00:42here what does this count as what does
2:00:46that mean an exponent that is an
2:00:48exponent so if this is a negative one
2:00:50then really what we can do
2:00:54we can move that as an exponent so this
2:00:56is Ln absolute value cosine
2:01:00X to the
2:01:051 tell me
2:01:10something how much is that equal to you
2:01:12should know it one over cosine very good
2:01:15this is one over cosine one / cosine is
2:01:17also known
2:01:19asant that's right
2:01:29so we get another identity for us what
2:01:34we know now is that the
2:01:36integral of tan X DX is equal to Ln
2:01:41absolute value see
2:01:43X plus C some people out there might be
2:01:47wondering wait a second
2:01:49um isn't this a problem because is if I
2:01:53plug
2:01:54in if I plug in Pi /
2:01:572 that's zero are you with me you should
2:02:00know your trig right if I plug in pi
2:02:02over two you get zero so wouldn't this
2:02:04be one over Z isn't that undefined and
2:02:07answer is yeah would be until you
2:02:09consider where this is coming from
2:02:11tangent is not even defined at pi/ 2
2:02:14it's continuous and it's defined between
2:02:16not including pi2 and pi2 remember those
2:02:18S curves that you get so we don't really
2:02:20have to worry about the pi/2 because you
2:02:21would even be able to integrate It
2:02:23Anyway does that make sense to you okay
2:02:26you might not have thought of that but
2:02:27there it is so we got another
2:02:31one so we got these ones now we got
2:02:34integral of tangent equals Ln absolute
2:02:36value see X plus C here's the the deal
2:02:39in this class once you prove something
2:02:42you don't have to keep redoing it so
2:02:45next time you see the integral of
2:02:46tangent or you see something that has a
2:02:48tangent in there don't reinvent the
2:02:50wheel don't you don't have to break it
2:02:52up into sign over cosine we just did it
2:02:54anything that we prove in class like
2:02:56this one you can use so now use that
2:02:58keep that in mind next time you do your
2:02:59homework you go oh okay how about the uh
2:03:02you know we're not going to do this one
2:03:03but how about the integral of x tan x^2
2:03:07DX well you're thinking uh substitution
2:03:11for the X now that's in your table
2:03:13you're good to go does that make sense
2:03:15makes a whole lot easier than having to
2:03:16go through this all the way over again
2:03:19and again you guys want to see one more
2:03:22did you like that one it's kind of
2:03:24interesting right it's kind of fun uh
2:03:26tell you what we'll
2:03:28do run out of room
2:03:31here I think we can do
2:03:41this this one's tricky it's it's really
2:03:44tricky it It's tricky because we use a
2:03:47trick here that I I wouldn't expect you
2:03:48to know I didn't know it until I saw it
2:03:50so if you're like well where's that
2:03:52coming from so was I the first time okay
2:03:54what we're going to do here is we're
2:03:55going to multiply by one in a really
2:03:58really unique way we're going to change
2:04:00the look of this thing and we're going
2:04:02to change the look so that we can use a
2:04:04substitution that actually makes sense
2:04:06because right now doing the one over
2:04:07coine thing we're going stuck in a right
2:04:09because if you do one over Co are you
2:04:11familiar with the one over cosine here
2:04:12that I'm talking about you do one over
2:04:14cosine you do a substitution for cosine
2:04:15to get the one over you it doesn't the
2:04:17sign doesn't eliminate so you'd be stuck
2:04:19in a right back and forth and back and
2:04:21forth It's like Catch 22 okay so we're
2:04:23going to do a nice little trick
2:04:25here instead of just having secant x
2:04:28what I'm going to
2:04:30do is I'm going to keep the secant
2:04:33x and I'm going to
2:04:37multiply by see X Plus tangent X on both
2:04:42the numerator and
2:04:46denominator if you're like Leonard what
2:04:48in the world are you doing it's a trick
2:04:50that we're going to use to create a
2:04:52substitution all right I know it looks
2:04:54weird it's like where how would you even
2:04:55come with that I didn't come with it
2:04:56myself honestly I read it out of the
2:04:58book I'm like oh man these tricky guys
2:05:00you son of a gun you did it well here's
2:05:03the deal and you're going to see it
2:05:04right away as soon as we distribute this
2:05:06numerator notice numerator gets
2:05:08distributed denominator does not so we
2:05:10have
2:05:16here see 2 X Plus SEC tangent uh sorry
2:05:21seing X tangent
2:05:35X you guys okay with the
2:05:38distribution yes
2:05:40no se Square see tangent here's why we
2:05:44did that I want you to look at what's
2:05:47the worst part of this what's making
2:05:49this harder than it needs to be
2:05:52the denominator is the denominator is
2:05:54the bad part here if we do a
2:05:56substitution for
2:05:57this what's the derivative of secant squ
2:06:02oh yeah what's derivative of
2:06:04tangent oh sorry I had these backwards
2:06:07but what's the derivative of
2:06:09secant that one what's the derivative of
2:06:13tangent so if we do it together we're
2:06:15going to get that whole piece out of it
2:06:16does that make sense we're inventing a
2:06:18substitution here it's a little awkward
2:06:20cuz yeah it's hard to see it it's wow
2:06:23how in the world but someone figured it
2:06:25out and we're just going to use it so if
2:06:27we do U equals the
2:06:29denominator SEC X Plus tangent X and we
2:06:35do a derivative again let's try this
2:06:39derivative of see X everybody what is
2:06:43that very good
2:06:51plus so we did this this is that
2:06:54part plus that's right C
2:07:03squ I'm going to do the substitution the
2:07:05different way some of you are familiar
2:07:07with this uh you look here you go okay
2:07:09DX I try to look for this stuff too do
2:07:11you guys see that this is this whole
2:07:13part right here yeah that's our
2:07:17substitution that's
2:07:19awesome I love this game
2:07:23so we do a substitution we have tell me
2:07:25what the integral is now what's on the
2:07:27numerator one yeah one one's or du sure
2:07:31over what what did we call this
2:07:34U and then this whole thing became that
2:07:37du so you either have du over U or you
2:07:39have one over U du either way is fine
2:07:42can we do the integral now can now it's
2:07:44a piece of
2:07:45cake Ln absolute value of U plus C no
2:07:50problem the only that we're going to do
2:07:52is what put the U Back yeah put it right
2:07:55back in there so what we know now is
2:07:57that the integral of
2:08:02secant x
2:08:05DX got it from there
2:08:08is Ln absolute value remember this was
2:08:12our U see X Plus tangent X
2:08:22plus C that's the reason why you didn't
2:08:25get a whole bunch of these things in
2:08:26your Calculus one class is because you
2:08:27can't do them you can't do them until
2:08:29you learn about what the integral of one
2:08:31over you is once you do that the whole
2:08:32world opens up for you as far as
2:08:34trigonometric integrals uh did I explain
2:08:36this well enough for you show hands if I
2:08:38did you're okay with that one what I'm
2:08:39going to do now I'm not going to prove
2:08:40all of them I'm just going to complete
2:08:42this list for you that way you have the
2:08:44entire thing okay so we just did tangent
2:08:47we just did secant there's a couple of
2:08:48those I want to give
2:08:49you so as add on to whatever list you
2:08:54have the following
2:09:08things first one we already did we know
2:09:10that the integral of tangent U
2:09:13du is Ln absolute value see
2:09:17U plus
2:09:20C is that's not the same thing as saying
2:09:22x uh you know it is the same the reason
2:09:24why they have it as you is to let you
2:09:26know that you can make a substitution
2:09:28change it into that so you can put it X
2:09:30or a u um actually all of these should
2:09:33probably be used I think it's that way
2:09:34in the integration table in your book or
2:09:36whatever you're using um my bad I just
2:09:38did them on top of my head so I used X
2:09:39because I'm used to it um use X or youu
2:09:42interchangeably the reason again why
2:09:44they're using U here is to let you know
2:09:46that a substitution is possible in them
2:09:48so don't forget about that other words
2:09:53one we didn't do we didn't do coent U
2:09:57but I hope that you understand it would
2:09:59be really really similar to tangent all
2:10:02you'd have is cosine over s you do
2:10:05practically the same thing and you're
2:10:06going to
2:10:13get
2:10:19this this one we did
2:10:30do this one we did do did right over
2:10:34there so this would be Ln absolute
2:10:37value see U plus tangent
2:10:41U plus
2:10:45C and last one again the uh gu see what
2:10:50I'm talking about about the cotangent
2:10:51would look really the proof would look
2:10:53really really similar to this one well
2:10:56the
2:10:58cosecant proof would look really really
2:11:00similar to this it's it's a it's similar
2:11:02to that
2:11:03one so we'd have Ln absolute
2:11:07value cose
2:11:09U minus Cent
2:11:12U and there's some similarities here uh
2:11:15instead of secant we get cosecant
2:11:17instead of tangent we get Cent the plus
2:11:18we get minus it's kind of like the
2:11:20reverse of all of my suppos if you
2:11:21wanted to call it that show fans feel
2:11:24okay with with these ones I'm not going
2:11:25to prove them uh the proofs are really
2:11:27similar so you can either do them on
2:11:28your own look them up I think some of
2:11:29them are exercises for you what I do
2:11:31want to do is two examples to illustrate
2:11:34how you would go about doing these would
2:11:35you like to see some yeah okay
2:11:46so you guys like tattoos right just get
2:11:50an arm tattoo
2:11:52it'd be the coolest thing like ever
2:11:56no can you rip your arm
2:11:59off then
2:12:01yes do
2:12:07it I won't give you extra credit but
2:12:09you'll be
2:12:14awesome just tell them it's some sort of
2:12:17like I don't know Japanese writing most
2:12:19people w't know the difference
2:12:21some says honor and
2:12:28strength I did have a student once who
2:12:31came to school I got to tell the story
2:12:33came to school like he wore the same
2:12:35shirt like I think three times in the
2:12:37class and it was Japanese writing he was
2:12:39not Japanese so hey what's your shirt me
2:12:41he goes oh I really don't know and so I
2:12:43had two Japanese students I'm like do
2:12:44you know what this he left class uh to
2:12:46use restroom something I said do you
2:12:47know what that shirt means like yeah I
2:12:48said what's it mean I don't want to tell
2:12:49you like no come on tell me
2:12:51it means I want a girlfriend so funny he
2:12:55had no idea
2:13:04classic that that was a funny
2:13:07day I never told him I never told him
2:13:09what I meant
2:13:11any I don't want to make fun of them
2:13:13that would be mean f
2:13:16of okay so let's start with our problem
2:13:18does it look exactly like something in
2:13:20our integration table uh yeah no it's
2:13:23got X's it's got an X squ there it
2:13:24doesn't look like that so if it doesn't
2:13:26look exactly the same you're going to do
2:13:28what sub let's do a substitution uh
2:13:31what's the appropriate thing to
2:13:32substitute for here good yeah it's
2:13:34usually the inside of something the
2:13:36derivative is always in the integral you
2:13:37don't want to pick the whole thing
2:13:39because then you're going to get a trig
2:13:40function that's not in there again so
2:13:42that would be a problem so typically try
2:13:44the small stuff first like X squ in this
2:13:46case so if we substitute for
2:13:51x^2 of course du = 2x
2:13:54DX du over
2:13:572x = DX well then we have an X still
2:14:01here depending on how you
2:14:03substitute see U du over 2x is it going
2:14:08to work out for us yeah X's are gone
2:14:11it's fantastic tell me what else we
2:14:14do 12 in Fr yeah
2:14:1812 see U du you look what our
2:14:22substitution did what it did was it
2:14:24matched this up with something that we
2:14:26just learned how to do you don't have to
2:14:27reinvent the wheel all the time that be
2:14:29ridiculous so as soon as you do your
2:14:31substitution now you start looking at
2:14:33your your table of integration you go oh
2:14:35well how much is that do you know how
2:14:36much that is you know what the integral
2:14:39of that is think I'm covering it up it's
2:14:42right
2:14:44here so we still have the 1/2 we've got
2:14:48Ln we've got absolute value sure see U
2:14:53plus tangent U plus C the reason why
2:14:58they use the U here is because you are
2:15:00probably going to use the use here
2:15:02that's a lot of use do use understand me
2:15:05joking uh anyway the last thing that you
2:15:08want to do though please please don't
2:15:10forget to substitute back in a lot of
2:15:12people when are doing this they either
2:15:14do us they go I'm done that was so hard
2:15:16oh finally done or they just put X's
2:15:18right off the bat you don't want to do
2:15:19that either okay so what goes here here
2:15:21and here yeah so our answer is going to
2:15:24be 12 Ln absolute
2:15:27value X2 +
2:15:32tangent X2 + C you guys are right with
2:15:37that one can you put that one2 of an
2:15:39exponent as well you can not such a big
2:15:42deal I don't really care about that it
2:15:43is it is an exponent it would be the
2:15:45square root over all this stuff all
2:15:47right uh not super necessary for you to
2:15:49do that good question though any other
2:15:50question questions before we go on you
2:15:53ready for one
2:15:56more does anybody else need
2:15:59this okay
2:16:14cool okay this one's going to bend your
2:16:16brain a little bit you going have to
2:16:17think about this one
2:16:39that's
2:16:47another wait a minute uh
2:16:51that hey that definitely doesn't look
2:16:52anything like anything in our
2:16:53integration table right away right so
2:16:55we're going to have to do a
2:16:56substitution let's check out the
2:16:58appropriate substitution should it be
2:17:022x no if I do that it's not going to get
2:17:04rid of any of this stuff should it be
2:17:06the numerator at all no because if I do
2:17:10this it's not going to touch this down
2:17:11here so probably it's going to be
2:17:13something down here now let's talk about
2:17:16the denominator if I pick the entire
2:17:19denominator
2:17:26let's try the derivative of that what's
2:17:29the derivative of one well that's
2:17:31fantastic so when we have that get rid
2:17:33of that one for
2:17:35sure can you do the derivative of s
2:17:39squared oh come on what rule is that
2:17:42rule it's a chain rule that's right do
2:17:44you remember that sin^2 X means sin x
2:17:47all of it squared we'll bring down the
2:17:49two we we leave the inside alone we'll
2:17:52multiply by the derivative of the inside
2:17:54are you with me yeah we'll bring down
2:17:57the
2:17:57two we'll leave the inside alone we'll
2:18:00subtract one from it and we'll multiply
2:18:02by the derivative of the
2:18:05inside I want to know if you're okay
2:18:07with 2 sinx cosine X do you not so
2:18:11much
2:18:13derivative of sin x^ 2 that's what this
2:18:18means you have to know that part okay if
2:18:20we do the Dera of this this is 2 sin x
2:18:24to the first Power time the
2:18:27derivative of sin x what's the D of sin
2:18:31x two this is
2:18:37meaningless X that's our appropriate
2:18:40derivative yes no I know I went fast on
2:18:42that but that's old stuff are we okay
2:18:44with it yeah
2:18:46okay now wait a second does that even
2:18:50help us
2:18:52no explain why well because s of 2x the
2:18:57trick identity for that is 2 X I love
2:19:00those identities don't you love those
2:19:02identities this identity right
2:19:04here this is two SX cosine X that's the
2:19:10identity it's double
2:19:12angle you got to know those identities
2:19:15if not you got to got to refresh your
2:19:17memory like now okay go home and start
2:19:20you guys like tattoos right just joking
2:19:23you have a lot of tattoos let's see what
2:19:26happens here okay I forgot the DX sorry
2:19:29see what happens um let's do the easy
2:19:31for what's this
2:19:33become
2:19:35mhm if this is you know what I'm going
2:19:37to do this in two steps I'm sorry I'll
2:19:40do it in two
2:19:44steps the first step that I've done here
2:19:47it's just the identity I've just changed
2:19:49this into this um if you don't know that
2:19:51identity by heart you can look it up
2:19:52later it is a true identity okay so this
2:19:55is true at this point I would do my
2:19:57substitution now check out what happens
2:20:00I have a 2 sinx cosin x DX that's
2:20:03exactly this piece so
2:20:06this is going to change into my
2:20:11du so I get
2:20:14du over
2:20:16U one over U du or du over U it's the
2:20:20same thing
2:20:22okay I don't want to have I don't want
2:20:23to go too fast on that one I need to
2:20:24make sure that you guys are with me uh
2:20:26do you are you comfortable with this
2:20:28identity now that I told you that it's
2:20:30true are you okay with picking this as
2:20:32your U if you are then the derivative is
2:20:352 sin x cosine X that's literally
2:20:37exactly what we have here so this piece
2:20:40becomes du this piece becomes U and
2:20:44that's a really really easy integral how
2:20:46much is that
2:20:48integral abute yep
2:20:56we got one more thing to do we're going
2:20:58to put the
2:21:00U so Final Answer Ln absolute value 1 +
2:21:06s^ 2
2:21:08X plus C it's actually technically not
2:21:11the final answer we're going to do one
2:21:12more thing with this one more thing I
2:21:15want to make sure you're okay with this
2:21:16so far though are you the last thing
2:21:18we're going to do is we're going to drop
2:21:19the absolute value
2:21:21do you know why because it can never be
2:21:24negative that's right it can't ever be
2:21:25negative this is always positive if you
2:21:28add one to something that's always
2:21:29positive it's always positive so the
2:21:31absolute values are kind of redundant
2:21:33you can do that if you know your
2:21:34argument's always positive so we could
2:21:37write this instead as just
2:21:41Ln this
2:21:44stuff plus C and that's probably how
2:21:47you'll see it
2:21:51we good that's it that's 6.1 we're done
2:21:54so we just started off our Ln uh we know
2:21:57how to take derivatives and integrals do
2:21:59I have any questions whatsoever how many
2:22:01people feel okay with with the idea you
2:22:04guys have questions now so now you know
2:22:06how to do derivatives with Ln yes just
2:22:09do one over inside times derivative
2:22:12inside integrals a little bit more labor
2:22:15intensive you have to do some
2:22:16substitutions especially with trig stuff
2:22:18you're going to get a lot of trig stuff
2:22:19in this class get you used to it learn
2:22:22those identities and learn how to use
2:22:23them okay so right now what we're going
2:22:26to start we're going to start section
2:22:276.2
2:22:28[Applause]

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