Derivative as a concept | Derivatives introduction | AP Calculus AB | Khan Academy — Transcript

Full transcript

1. 0:00you are likely already familiar with the
2. 0:02idea of a slope of a line if you're not
3. 0:04I encourage you to review it on Con
4. 0:06Academy but all it is it's describing
5. 0:09the rate of change of a vertical
6. 0:11variable with respect to a horizontal
7. 0:13variable so for example here I have our
8. 0:16classic y AIS in the vertical Direction
9. 0:18and xaxis in the horizontal Direction
10. 0:20and if I wanted to figure out the slope
11. 0:22of this line I could pick two points say
12. 0:25that point and that point I could say
13. 0:27okay from this point to this point what
14. 0:29is my change in X well my change in X
15. 0:32would be this distance right over here
16. 0:34change in X the Greek letter Delta this
17. 0:38triangle here it's just shorthand for
18. 0:40change so change in X and I could also
19. 0:43calculate the change in y so this point
20. 0:48going up to that point our change in y
21. 0:50would be this right over here our change
22. 0:54in y and then we would Define slope or
23. 0:58we have defined slope as change in y
24. 1:01over change in X so slope is equal to
25. 1:05the rate of change of our vertical
26. 1:07variable over the rate of change of our
27. 1:09horizontal variable it's sometimes
28. 1:11described as rise over run and for any
29. 1:16line It's associated with a slope
30. 1:18because it has a constant rate of change
31. 1:21if you took any two points on this line
32. 1:24no matter how far apart or no matter how
33. 1:26close together anywhere they sit on the
34. 1:28line if you were to do this calculation
35. 1:31you would get the same slope that's what
36. 1:34makes it a line but what's fascinating
37. 1:37about calculus is we're going to build
38. 1:39the tools so that we can think about the
39. 1:41rate of change not just of a line which
40. 1:43we've called slope in the past we can
41. 1:45think about the rate of change the
42. 1:48instantaneous rate of change of a curve
43. 1:51of something whose rate of change is
44. 1:54possibly constantly changing so for
45. 1:57example here's a curve where the rate of
46. 2:00change of Y with respect to X is
47. 2:03constantly changing even if we wanted to
48. 2:06use our traditional tools if we said
49. 2:08okay we can calculate the average rate
50. 2:10of change let's say between this point
51. 2:13and this point well what would it be
52. 2:16well the average rate of change between
53. 2:17this point and this point would be the
54. 2:18slope of the line that connects them so
55. 2:21it be the slope of this line of the
56. 2:23secant line but if we pick two different
57. 2:26points if we pick this point and this
58. 2:27point the average rate of change between
59. 2:29those points all of a sudden looks quite
60. 2:31different it looks like it has a higher
61. 2:34slope so even when we take the slopes
62. 2:37between two points on the line the
63. 2:39secant lines you can see that those
64. 2:42slopes are changing but what if we
65. 2:44wanted to ask ourselves an even more
66. 2:46interesting question what is the
67. 2:48instantaneous rate of change at a point
68. 2:52so for example how fast is y changing
69. 2:55with respect to X exactly at that point
70. 2:58exactly when X X is equal to that value
71. 3:01let's call it
72. 3:02X1 well one way you could think about it
73. 3:05is what if we could draw a tangent line
74. 3:08to this point a line that just touches
75. 3:10the graph right over there and we can
76. 3:12calculate the slope of that line well
77. 3:15that should be the rate of change at
78. 3:18that point the instantaneous rate of
79. 3:20change so in this case the tangent line
80. 3:24might look something like that if we
81. 3:27know the slope of this
82. 3:30well then we could say that that's the
83. 3:31instantaneous rate of change at that
84. 3:33point why do I say instantaneous rate of
85. 3:36change well think about the video on the
86. 3:39sprinters the usin bolt example if we
87. 3:42wanted to figure out the speed of Usain
88. 3:44Bolt at a given instant well maybe this
89. 3:47describes his position with respect to
90. 3:49time if y was position and X is time
91. 3:52usually you would see T is time but
92. 3:54let's say x is time so then if we're
93. 3:55talking about right at this time we're
94. 3:58talking about the instant instantaneous
95. 4:01rate and this idea is the central idea
96. 4:04of differential calculus and it's known
97. 4:07as a
98. 4:08derivative the slope of the tangent line
99. 4:11which you could also view as the
100. 4:13instantaneous rate of change I'm putting
101. 4:16exclamation mark because it's so
102. 4:18conceptually important here so how can
103. 4:20we denote a derivative one way is known
104. 4:24as Liv Net's notation and livet is one
105. 4:27of the fathers of calculus along with
106. 4:29Isaac Newton
107. 4:30and his notation you would denote the
108. 4:32slope of the tangent line as equaling dy
109. 4:37over
110. 4:38DX now why do I like this notation
111. 4:42because it really comes from this idea
112. 4:44of a slope which is change in y over
113. 4:46change in X as you'll see in future
114. 4:49videos one way to think about the slope
115. 4:51of the tangent line is well let's
116. 4:53calculate the slope of secant lines
117. 4:55let's say between that point and that
118. 4:56point but then let's get even closer say
119. 4:58that point and that point and then let's
120. 5:00gets even closer and that point and that
121. 5:01point and then let's get even closer and
122. 5:03let's see what happens as the change in
123. 5:06X approaches zero and so using these D's
124. 5:10instead of Deltas this was liet's way of
125. 5:13saying hey what happens if my changes in
126. 5:16say x become close to zero so this idea
127. 5:20this is known as sometimes differential
128. 5:22notation liances notation is instead of
129. 5:25just change in y over change in x super
130. 5:28small changes in y for a super small
131. 5:31change in X especially as the change in
132. 5:34X approaches zero and as you will see
133. 5:36that is how we will calculate the
134. 5:38derivative now there's other
135. 5:41notations if this curve is described as
136. 5:44Y is equal to F ofx the slope of the
137. 5:48tangent line at that point could be
138. 5:50denoted as equaling F Prime of X1 so
139. 5:57this notation it takes a little bit of
140. 5:58time getting used to the lrange notation
141. 6:01it's saying F Prime is representing the
142. 6:03derivative it's telling us the slope of
143. 6:06the tangent line for a given point so if
144. 6:10you input an X into this function into F
145. 6:13you're getting the corresponding y value
146. 6:16if you input an X into F Prime you're
147. 6:19getting the slope of the tangent line at
148. 6:23that point now another notation that
149. 6:26you'll see less likely in a Calculus
150. 6:28class but you might see in a physics
151. 6:30class is the notation y with a DOT over
152. 6:34it so you could write this as y with a
153. 6:37DOT over it which also denotes the
154. 6:39derivative you might also see y Prime
155. 6:42this would be more common in a math
156. 6:45class now as we March forward in our
157. 6:48calculus Adventure we will build the
158. 6:50tools to actually calculate these things
159. 6:52and if you're already familiar with
160. 6:54limits they will be very useful as you
161. 6:55can imagine because we're really going
162. 6:56to be taking the limit of our change in
163. 6:59y over change in X as our change in X
164. 7:02approaches zero and we're not just going
165. 7:05to be able to figure it out for a point
166. 7:07we're going to be able to figure out
167. 7:08General equations that describe the
168. 7:10derivative for any given point so be
169. 7:14very very excited