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Chain Rule For Finding Derivatives — Transcript

by The Organic Chemistry Tutor · 2,453 words · 359 segments · language en · Watch on YouTube

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0:00let's move on to the chain rule we're
0:02going to cover a lot of examples the
0:05first Formula you need to be familiar
0:06with is the derivative of the composite
0:09function f of g
0:11ofx a composite function is one where
0:14you have one function inside of another
0:18notice that g is inside of f which makes
0:20it a composite function so the first
0:23thing you need to do is differentiate
0:25the outside portion of the function that
0:26is f and you need to keep the inside the
0:29same
0:30and then multiply it by the derivative
0:32of the inside that's the main idea
0:34behind the chain rule if you follow this
0:37process you're going to get the answer
0:38right so let's say for
0:41example if we have a function U raised
0:44to the N where U is another function in
0:46terms of X using the chain Rule and the
0:49power rule combined it's going to be n *
0:53U you have to keep that the same raised
0:55to the N minus one times the derivative
0:58of what's on the inside that's the
1:00general power rule formula with the
1:01chain rule combine so never forget to
1:04multiply by the derivative of the inside
1:09function so let's use an example let's
1:12say if we want to find the
1:18derivative of 5x +
1:213 raised to the 4th power so the first
1:25thing we're going to do is we're going
1:26to move the constant I mean the exponent
1:28to the front so it's going to be four
1:31and then keep the inside stuff the same
1:33* 5x + 3 subtract the exponent by 1 4 -
1:381 is 3 and then multiply by the
1:40derivative of the
1:42inside the inside function is four it's
1:455x + 3 the derivative of 5x + 3 is just
1:495 and so that's the answer we can
1:52multiply four and 5 that's going to give
1:54us 20 so it's 20 * 5x + 3 ra the thir
2:00power so that's the final answer fully
2:04simplified now let's work on some more
2:07examples find the derivative of x^2 - 3x
2:12raised to the 5th
2:14power so first let's bring down to five
2:17so it's going to be five and then keep
2:20the inside function the
2:22same and then subtract the exponent by 1
2:24so this is four and then multiply by the
2:27derivative of the inside the derivative
2:29of x^2 - 3x is 2x - 3 and so that's the
2:34answer once you get used to the process
2:36it's not that
2:37bad here's another example that you can
2:39try find the derivative of s of
2:466X the derivative of the outside part of
2:48the function s is cosine and you got to
2:51keep the inside function the same then
2:54you multiply by the derivative of the
2:56inside function the derivative of 6X is
2:596 so the answer is simply 6 cosine
3:096X now what is the
3:12derivative of cosine
3:15x^2 so first differentiate the outside
3:18part of the function cosine the
3:21derivative of cosine is negative sign
3:24now the inside part of the function has
3:26to remain the same that is the angle of
3:28cosine so it's going to be x^2 and then
3:32differentiate the inside function x^2
3:36which is
3:372x so basically you're working away from
3:40the outside towards the inside the final
3:42answer is
3:44-2X sin
3:49X2 find the derivative of tangent X CU
3:54so first let's differentiate tangent the
3:57derivative of tangent is secant squ and
4:01the inside function has to remain the
4:03same next differentiate the inside
4:05function X Cub so that's
4:083x^2 and it's always multiplication so
4:11it's going to be
4:123x^2 secant 2 x
4:19Cub as you can see it's not that
4:23bad here's another problem what is the
4:25derivative of secant 4X
4:29the derivative of secant 4X is going to
4:32be secant tangent that's the derivative
4:34of
4:35secant now the inside function has to
4:38remain the same for secant and
4:40tangent next we need to differentiate 4X
4:44so it's just going to be time4 and so
4:47that's the
4:49solution what is the
4:51derivative of Ln X raised to the 7th
4:56power try that
4:58problem so this is going to be
5:01S keep the inside part the same and then
5:04subtract the exponent by one so 7 - 1 is
5:086 now we got to multiply by the
5:10derivative of the inside function the
5:13derivative of Ln X is simply 1 /x so the
5:16final answer is 7 Ln X raised to 6 power
5:21/
5:28X What is the dtive of
5:32theun of XB -
5:357 take a minute and work on that example
5:38the first thing I would do is rewrite it
5:40so this is the same as X Cub - 7 raised
5:44to the
5:4712 and so that's going to be equal
5:50to2 we got to bring the exponent to the
5:53front keep the inside function the same
5:56and then subtract the exponent by one 1
5:5912 - 1 which is 12 - 2 2 that's a half
6:04and then we got to multiply by the
6:06derivative of the
6:07inside the derivative of x Cub - 7 is
6:10simply
6:123x^2 so we could bring this back to the
6:14bottom since it has a negative exponent
6:16so it's 3x^2 / we have a two on the
6:20bottom 2 XB - 7 and now the exponent is
6:25going to change from negative half to
6:27positive half and now we could put it
6:30back in its radical form so it's 3x^2 /
6:332 < TK XB - 7 and so that's the final
6:39answer for this
6:41problem find the
6:44derivative of 1
6:46/ x^2 + 8 raised to the 3
6:52power so first let's rewrite the
6:56expression let's bring the variables to
6:58the top so this is is x^2 + 8 raed Theus
7:033 and now we can use the chain
7:06rule combined with the power rule let's
7:08move the3 to the front and let's keep
7:11the inside function let's rewrite it
7:13exactly the way we see it and then let's
7:15subtract this by 1 -3 - 1
7:19is4 and now let's multiply by the
7:21derivative of the inside function which
7:23is
7:242x so now let's take this term move it
7:26back to the bottom so we have
7:29-3 * 2x which is -6x on top and on the
7:35bottom it's x^2 + 8 raised to the 4th
7:39power and so that's all we need to do
7:41for this
7:42problem so for some examples you need to
7:45rewrite it before you find the
7:51derivative now what if we have a trig
7:53function inside another trig
7:57function find the Der
7:59of this uh
8:04function so first we need to
8:06differentiate the outside function s the
8:09derivative of s is
8:11cosine now what's inside of cosine since
8:15cosine came from s everything inside of
8:18s is going to be the stuff inside of
8:21cosine so that's tangent X
8:244th now let's move on to tangent the
8:27derivative of tangent is secant squ
8:30and the stuff that's inside of tangent
8:32is going to be the stuff that's inside
8:34of secant squ so that's x
8:364 and then we got to move on further
8:39towards the inside the derivative of x
8:414th is 4X cub and so that's the
8:46answer how about this example find the
8:49derivative of s raised to the 5th
8:53power
8:56tangent
8:57cosine X cub
9:01the first thing I would do is rewrite
9:02the
9:04expression so this is
9:06s
9:08tangent
9:10cosine XB all rais the 5th power so
9:14first let's deal with the exponents
9:15let's use the power rule so we got to
9:18bring the five to the front and keep
9:19everything inside the same so always
9:23start with the outside portion of the
9:25function which is going to be the
9:26exponents in this case and then subtract
9:29the exponent by one so it's going to be
9:31four now let's work our way towards the
9:34inside so we got a differentiate sign
9:36the derivative of s is cosine and the
9:39stuff inside of s is tan cosine X Cub so
9:43we got to put that
9:50here now let's move on to tangent the
9:53derivative of tangent is secant squ and
9:56the stuff inside of tan is cosine X cub
10:01so hopefully you're seeing a pattern
10:03with the way we're finding the
10:05derivative of everything so now that
10:08we're done with Tangent let's move on to
10:10cosine the derivative of cosine is
10:13negative sign and the angle of cosine is
10:16X Cub so that's going to be the same
10:17here and then let's move on to X Cub the
10:20derivative of x Cub is
10:233x2 so that's the answer and then you
10:26can combine terms you can multiply five
10:28and three to get 15
10:31what is the
10:34derivative of cosine raised to the 7th
10:37power of
10:40s of secant
10:43x^2 so try that problem so first let's
10:46rewrite it as
10:48cosine of s of secant x^2 and let's put
10:53the seven and it's exponent position so
10:56let's use the power rule let's bring the
10:58seven to the front
10:59and let's keep everything on the inside
11:02the
11:07same and then subtract the exponent by
11:09one so this is going to be six now let's
11:12find the derivative of cosine let's work
11:14our way towards the inside the
11:16derivative of cosine is negative sign
11:18and the stuff inside of that is s
11:23secant
11:27X2 so now we got to find the Der ative
11:31of s the derivative of
11:34s is
11:37cosine and the stuff inside of s is
11:39secant
11:41x^2 so now we got to find the derivative
11:44of
11:44secant so that's secant tangent so it's
11:48going to be secant x^2 tangent
11:52x^2 and the derivative of x^2 is
11:562x so whenever you have multiple
11:58composite functions
12:00just work your way from the outside
12:02towards the inside and everything is
12:03multiplied by each
12:05other and then when you're done simply
12:07collect terms so we have a seven a 2X
12:10and a negative so you can move that to
12:12the front and write it as -4x if you
12:14want
12:16to here's the next
12:18problem find the derivative of x
12:22Cub * 4x + 5 raised to the 4th power
12:30so what we have here is a product rule
12:33we could say that this is f and this is
12:35G and for G we have to use the quotient
12:39rule I mean not the quotient rule but
12:40the chain rule so using the product rule
12:43we need to differentiate the first part
12:45F the derivative of the first part is
12:483x^2 and we need to keep the second part
12:51the same so we just have to rewrite G
12:54plus now we need to keep the first part
12:56the same now for the second part we need
12:58to use the chain room so let's bring the
13:00four to the front let's keep the inside
13:02stuff the same and subtract the four by
13:05one then multiply by the derivative of
13:07the inside function which is 4x + 5 the
13:10derivative of that is
13:12four so now let's simplify so the first
13:15part we don't really need to change
13:17anything we can just leave it like this
13:20now for the second part we can multiply
13:22four and four that's 16 so this is 16 x
13:26Cub * 4x + 5
13:29raised to the thir power you can leave
13:31your answer like this or if you want to
13:34you can take out the GCF we can take out
13:36an x
13:38s and we could take out three 4x+ 5S or
13:43basically 4x + 5 to the third
13:47power so this is gone we took three of
13:50these one is left over and we have a a
13:52three left so this is going to be a 3 *
13:564x + 5
13:59and this is gone we took out all three
14:01of
14:02these well there's an X left over and
14:05it's 16 so plus 16 x because we took out
14:08an X squ now what we can do is basically
14:12simplify what we have
14:16here so this is x^2 4x + 5 to the 3
14:22power and let's distribute the three so
14:243 * 4X is 12x 3 * 5 is 15
14:29+
14:3116x now let's add 12x and 16x so the
14:35final
14:36answer is x^2 * 4x + 5 to the 3
14:40power and then 28x + 15 so that's the
14:48solution let's try one more example
14:52let's find the
14:53derivative of 2x - 3 / 4 + 5x
15:00X raised to the 4th
15:04power so this is a chain Rule and
15:07quotient rule problem so first we need
15:09to bring the exponent down and keep the
15:11stuff on the inside the
15:14same and then subtract the exponent by
15:17one next we need to multiply by
15:20derivative of the inside so that's when
15:22we have to use a quotient
15:24rule so f is 2x - 3 and G
15:29is 4 +
15:315x frime is 2 G Prime is 4 and the
15:36formula for the quotient rule is it's g
15:39f Prime minus FG Prime over G ^2 so G is
15:454 +
15:465x F Prime is
15:512 and
15:53F is 2x -
15:563 G Prime is 4 / g^ 2 which is 4 +
16:035x^2 so as you can see this is a long
16:12problem so let's simplify what we have
16:15on the right side if we distribute the
16:18two it's going to be
16:208 + 10
16:22x and if we distribute the four it's
16:25going to be 8X and then * -3 * 4 that's
16:34pos2
16:35/ 4 +
16:395x^2 and then this part is going to be
16:42the
16:52same so let's simplify the numerator so
16:56first we can add 8 and 12
16:59so that's going to be 20 and then 10 x -
17:028X that's positive 2x / 4 +
17:105x^2 now this thing we could distribute
17:13the three to the numerator and the
17:16denominator so we can write it like this
17:17if you want 2x - 3 Cub / 4 + 5x^2 the
17:24reason why I did that is because I can
17:26now combine these two terms this
17:28supposed to be Cube not
17:36squared so now what we have is 4 * 2x -
17:413 the 3 power and I'm going to take out
17:43a two here so if I take out a two it's
17:46going to be a x +
17:4810 over 4 + 5x raised to the fifth power
17:543 + 2 is 5 so the final answer is eight
17:594 * 2 is
18:018 * x +
18:0310 * 2x - 3 to the 3 power / 4 + 5x to
18:12the 5th
18:13power and so this is the final answer

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