What is a derivative? — Transcript

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1. 0:00is you know what is a derivative? Well,
2. 0:02basically what a derivative is, it's a
3. 0:04way of finding the instantaneous rate of
4. 0:06change. And when you hear the word rate
5. 0:08of change, what that means is referring
6. 0:10to the slope. So, let me draw a diagram
7. 0:12here for us. So, basically what we have
8. 0:14is something that looks like this. Okay,
9. 0:17we have a curve for example, right? And
10. 0:20we have two points on the curve. Let's
11. 0:22just say right here and right here. Now,
12. 0:24if we go down to the x-axis, okay, this
13. 0:27is going to be x. This distance right
14. 0:28here is x. And if we go up to our
15. 0:30function, this point here is going to be
16. 0:33x comma f ofx. Okay, so when you put
17. 0:36that x value into your function, you get
18. 0:38out the y-value or the f ofx value.
19. 0:40Right? Now, if we go a little bit
20. 0:42further to the right, okay, let's just
21. 0:44call this distance h. Okay, so you're
22. 0:47with me so far. That means over here,
23. 0:49we're going x plus h. Okay, x plus h.
24. 0:52And if we go up to this point on our uh
25. 0:55graph right here, we've got x plus h. If
26. 0:57we put that into the function, the
27. 0:59output or the y value is f ofx plus h.
28. 1:03Okay, so all we've done is we've just
29. 1:05written this in a real generic way, but
30. 1:07what we have is we have two points on
31. 1:09our curve. And now what we're going to
32. 1:10do is we're going to find the slope,
33. 1:14okay, between those two points. Okay?
34. 1:16And when you find the slope, remember
35. 1:18your slope formula from algebra. It's y2
36. 1:20- y1 over x2 - x1, right? That's our
37. 1:24slope formula. In this case, it's going
38. 1:26to be f ofx + h.
39. 1:31Okay, that's that y value minus f ofx,
40. 1:34right? / x2 - x1. So that's going to be
41. 1:37x + h.
42. 1:39Okay, min - x. And you can see here that
43. 1:42the uh
44. 1:45x plus h - x, right? You can see that
45. 1:48the x's are cancelling and we just have
46. 1:49h in the denominator, right? But now
47. 1:51let's take this a step further. So what
48. 1:53we're doing right now is we're actually
49. 1:54finding the slope between these two
50. 1:56points which is the average rate of
51. 1:58change. Okay, it's this average slope.
52. 2:01But you can see that anywhere along this
53. 2:02graph, the slope is constantly changing.
54. 2:04Like here, okay, it's at an angle like
55. 2:06so. Here it's a little bit less steep.
56. 2:08Here it's even less steep. Here it's
57. 2:09even less steep. Here it's even turning
58. 2:11around and going the other direction.
59. 2:12Right? So what we're doing is we're
60. 2:14finding a formula for the instantaneous
61. 2:17rate of change. Okay? The slope at any
62. 2:20point along this curve. And the way we
63. 2:22do that is we introduce limits. And it's
64. 2:25the limit as h approaches zero. What
65. 2:29that means is as this distance right
66. 2:31here gets smaller and smaller and
67. 2:32smaller, this point here in effect is
68. 2:36moving along this curve until these two
69. 2:39points are right on top of one another
70. 2:43and that distance is zero. And so what
71. 2:45we're getting now is we're getting the
72. 2:48instantaneous rate of change instead of
73. 2:50this average rate of change. So what
74. 2:52you're seeing right here, and let me
75. 2:53just rewrite it a little bit, is this is
76. 2:55the limit as h approaches zero of f(x) +
77. 2:59h - fx all / h. This is referred to as
78. 3:04the difference quotient. Difference,
79. 3:06remember, is subtracting. Quotient is
80. 3:08dividing. So altogether, they call this
81. 3:10the difference quotient. But when you
82. 3:12get to the end of pre-calculus,
83. 3:14beginning of calculus, they start to
84. 3:16refer to this as fprime of x, which is
85. 3:19the derivative. So don't get confused by
86. 3:22the uh the word derivative. It's really
87. 3:24just a formula for the slope of the
88. 3:26tangent line. Okay? The line that just
89. 3:28barely touches that graph at that one
90. 3:29point. And it gives us the instantaneous
91. 3:31rate of change. So let's get into uh a
92. 3:35basic example. Now say for example, your
93. 3:37function is like this. f ofx=
94. 3:40x^2. Okay, so we've grown up uh as we've
95. 3:43gone along in math and we know what f
96. 3:45ofx= x^2 looks like. It's just a
97. 3:47parabola like so. Right? But now what
98. 3:50we're going to do is we're going to use
99. 3:51our derivative. We're going to use our
100. 3:52difference quotient to find the
101. 3:54derivative. Okay, which is a formula for
102. 3:56the slope of the tangent line at any
103. 4:00point along this graph. So what we're
104. 4:01going to do is let's go ahead and do
105. 4:02these substitutions here. We're going to
106. 4:04put in x + h, okay, squared minus f ofx,
107. 4:10which is the original function. That's
108. 4:11x^2 all / h. So if we simplify this, we
109. 4:14get x^2 + 2 hx + h 2 - x^2 all / h. And
110. 4:21it's the limit as h is approaching zero.
111. 4:24Right? Now notice that the x^2s are
112. 4:26cancel each other out. And then here we
113. 4:28can factor out an h which gives us 2x +
114. 4:31h all / h. So those h's cancel. And
115. 4:34again, remember it's the limit as h
116. 4:35approaches zero. If we put zero in for
117. 4:37h, we're left with 2x. And this is our
118. 4:40formula
119. 4:43for our derivative, which remember the
120. 4:44derivative is just a formula for the
121. 4:46slope of the tangent line. So a simple
122. 4:48example would be say we were interested
123. 4:50in finding the slope at the point 2
124. 4:544. Okay, right there. Okay, so we're
125. 4:56trying to find the slope of that tangent
126. 4:58line. Well, if we put in that x
127. 5:00coordinate 2, okay, in for x here,
128. 5:02that's going to be frime of 2, which is
129. 5:05equal to 2 * 2, which is equal to 4. And
130. 5:08again, remember this is a formula for
131. 5:10our slope. So now we know that the slope
132. 5:12is 4 and it goes through the point 24.
133. 5:15Let's write an equation of the tangent
134. 5:18line. So that's going to be y - y1 = m *
135. 5:23x - x1. We're just going to use the
136. 5:24point slope form. So this is going to be
137. 5:26y - 4 = the slope which is also 4 * x -
138. 5:31x coordinate of the point which is 2.
139. 5:33And that's an equation for our tangent
140. 5:36line. Okay? Like so.
141. 5:40Now let's get into another example about
142. 5:42you know we talked about how to find it.
143. 5:44We talked about what it is and another
144. 5:46example of why it's useful. So you
145. 5:49probably remember from algebra 2 and uh
146. 5:51pre-calculus when you were graphing
147. 5:53polomials. So take for example this
148. 5:55polomial here f ofx equ= uh x cubed - 4x
149. 6:02right so one way we would graph this is
150. 6:04we'd say all right let's see if we can
151. 6:06factor this and find the zeros remember
152. 6:08the zeros are just the
153. 6:10x intercepts so if we factor this
154. 6:12further we get x + 2 x -2 bring down the
155. 6:16x if we set all these factors equal to
156. 6:19zero you can see what we're getting is
157. 6:21we're getting zero we're getting
158. 6:24positive2 if we set that one to zero.
159. 6:26And we're getting -2 if we set this
160. 6:28group equal to zero. And we know the
161. 6:30leading coefficient's positive. Okay,
162. 6:32which tells us that the end behavior is
163. 6:33going up to the right. And we know it's
164. 6:36an odd degree. So that means it's going
165. 6:37the opposite direction to the left. And
166. 6:39we know the graph goes something like
167. 6:41that through those points. But the
168. 6:43question that students often times have
169. 6:46is when you go through these points,
170. 6:47like how high does that graph uh go
171. 6:50before it turns around? Does it go way
172. 6:52up here and then come back down here and
173. 6:53go way back down here and or does it
174. 6:55just go a little bit past the x-axis?
175. 6:57You know, how do we find that out? How
176. 6:59do we find those turning points? Well,
177. 7:01what's interesting is is that when the
178. 7:02graph bends, when it turns, okay, say
179. 7:04for example, like this. What's the slope
180. 7:06of the tangent line at that point where
181. 7:08it turns? Well, you can see it's a
182. 7:11horizontal line. So, the slope of that
183. 7:13tangent line is zero. So what we're
184. 7:15going to do is we're going to find the
185. 7:16derivative of our equation and we're
186. 7:17going to set that derivative equal to
187. 7:19zero so we can find the points on the
188. 7:21graph where the slope of the tangent
189. 7:23line is equal to zero. Okay, so we're
190. 7:25going to use our difference quotient
191. 7:26again, our derivative formula right
192. 7:27here. And so let's do that. So we've got
193. 7:30x + h cubed - 4 * x + hus
194. 7:37fx, which is original function. So
195. 7:39that's going to be xqub - 4x all / h.
196. 7:44Okay. Now if I simplify this, this comes
197. 7:46out to x cub + 3x^2h + 3xh ^2. I'm using
198. 7:52the binomial expansion theorem here.
199. 7:54Distribute the -4. So that's -4x - 4h.
200. 7:58Distribute the negative here. That gives
201. 8:00us x cub + 4x. And this is all divided
202. 8:04by h. Okay. And you can see we get a lot
203. 8:06of cancellation. The x cubes are
204. 8:08cancelling. The 4 x's are cancelling.
205. 8:11And then if we factor out an h out of
206. 8:13what remains here, we get 3x^2 + 3xh +
207. 8:18h2 - 4 all / h. You can see those h's
208. 8:23are cancelling. But remember, it's the
209. 8:25limit as h approaches zero. Right? So if
210. 8:28we put zero in now for h, this whole
211. 8:30term is going to go to zero. This whole
212. 8:32term is going to go to zero. and we're
213. 8:33left with 3x^2 minus 4 is our
214. 8:38derivative. Okay, that's a formula for
215. 8:40our slope. But what we want to do is we
216. 8:42want to find out where that slope is
217. 8:43zero. Okay, that's where the graph is,
218. 8:45you know, turning around or bending. And
219. 8:47so what we're going to do is we're going
220. 8:48to add four to both sides. Okay, so
221. 8:50that's 3x^2. We're going to divide both
222. 8:51sides by 3. So that's 4/3. We're going
223. 8:54to take the square root of both sides.
224. 8:56That gives us plus or minus, and I did
225. 8:58this on the calculator for us. It's
226. 8:591.15.
227. 9:01Okay. Now, if we put 1.15 back into our
228. 9:05original equation, what we get is 1.15,
229. 9:09comma,
230. 9:10uh, -3.08.
231. 9:12And if we put in negative 1.15, we get
232. 9:15positive 3.08.
233. 9:18So, those are our turning points. So, if
234. 9:19we graph this, you can see at 1.15,
235. 9:22we're going to be up here at
236. 9:24uh, let's see, we're going to be at
237. 9:263.08. So, it's going to be right about
238. 9:28there. And if we're at positive 1.5,
239. 9:31we're going to be right about there. So
240. 9:33that gives us a better idea, okay, of
241. 9:37how high the graph is going and how low
242. 9:40the graph is going. Otherwise, we're
243. 9:41just kind of guessing. We don't really
244. 9:43know where those extra are, those
245. 9:45maximums and those minimums, right? So
246. 9:47that's a a key point. And again, you can
247. 9:49see where that graph is bending or
248. 9:52turning or changing direction. That's
249. 9:53where the slope of the tangent line is
250. 9:55equal to zero. So key points to remember
251. 9:57when you're uh learning about
252. 10:00derivatives is you know what is a
253. 10:01derivative? It's really a formula for
254. 10:03the slope. How do we find the slope? We
255. 10:05take these two points, okay, and we use
256. 10:08our slope formula. But we take it a step
257. 10:10further to go from the average rate of
258. 10:12change to the instantaneous rate of
259. 10:15change by using limits. And the limit as
260. 10:18h gets smaller and smaller, it
261. 10:19approaches zero. And then from there
262. 10:22once you have the derivative you have a
263. 10:23formula for the slope of the tangent
264. 10:25line and you can you know use it in some
265. 10:27of these applications like we talked
266. 10:28about today. So I hope this video helped
267. 10:30you understand what is a derivative, how
268. 10:32is it useful you know and how to use it.
269. 10:35Subscribe to the channel, check out more
270. 10:36math tutoring videos on my YouTube
271. 10:38channel, Mario's Math Tutoring and I
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