1
00:00:00,012 --> 00:00:08,200
[MUSIC] Here I've graphed some function. 
Notice what happens if I zoom in on one 

2
00:00:08,200 --> 00:00:10,946
little piece of the graph, 
right? 

3
00:00:10,946 --> 00:00:18,594
If I just focus on one part of the graph, 
well, that little bit of graph sort of 

4
00:00:18,594 --> 00:00:25,505
looks like part of a straight line. 
Another way to think about this is as a 

5
00:00:25,505 --> 00:00:30,461
limit of secant lines. 
By secant line, I mean, I'm going to pick 

6
00:00:30,461 --> 00:00:36,570
two points on the graph and I'm going to 
put a line, the secant line, through 

7
00:00:36,570 --> 00:00:39,378
those two points. 
But, I can do better. 

8
00:00:39,378 --> 00:00:44,312
By taking this point and moving it closer 
to a, this red line is going to be a 

9
00:00:44,312 --> 00:00:48,311
better approximation to the orange curve 
near the point a. 

10
00:00:48,311 --> 00:00:53,347
So, instead of putting it through those 
two points, let me put the secant line 

11
00:00:53,347 --> 00:00:58,532
through, say, these two points, this 
point at a and this point that's nearby. 

12
00:00:58,532 --> 00:01:04,147
And now, the line segment through those 
two points is a much better approximation 

13
00:01:04,147 --> 00:01:08,466
of the orange curve. 
And I can do better and better by taking 

14
00:01:08,466 --> 00:01:13,832
a limit, by putting those two points 
closer and closer together, I can get my 

15
00:01:13,832 --> 00:01:19,391
secant line to be a better and better 
approximation to the orange curve near 

16
00:01:19,391 --> 00:01:22,758
the point a. 
What we're really calculating here is a 

17
00:01:22,758 --> 00:01:25,537
limit. 
So here, I've got an input a and an input 

18
00:01:25,537 --> 00:01:29,932
a+h, and here are the corresponding 
points on the graph of the function. 

19
00:01:29,932 --> 00:01:33,122
I'm going to put a secant line through 
those points. 

20
00:01:33,122 --> 00:01:38,172
What I want to know is what's the slope 
of that secant line because I'm going to 

21
00:01:38,172 --> 00:01:43,347
take the limit as this a+h point is moved 
closer to a as h goes to zero in other 

22
00:01:43,347 --> 00:01:48,497
words, and that'll make this secant line 
move closer and closer to the tangent 

23
00:01:48,497 --> 00:01:50,136
line, 
to the curve. 

24
00:01:50,136 --> 00:01:53,889
Okay, so what's the slope of the secant 
line? 

25
00:01:53,889 --> 00:01:58,431
Well, the rise is f(a)+h-f(a). 
The run is h. 

26
00:01:58,431 --> 00:02:01,771
So, the slope of that secant line is 
this, f(a+h)-f(a)/h. 

27
00:02:04,842 --> 00:02:10,137
If I take the limit as h goes to 0, I get 
this: the limit as h goes to 0 of this 

28
00:02:10,137 --> 00:02:13,507
slope. 
We've seen this, this is the derivative 

29
00:02:13,507 --> 00:02:18,442
of the function at the point a. 
Let's find the equation of the tangent 

30
00:02:18,442 --> 00:02:22,182
line in a concrete example. 
Here's a graph of y=x^2. 

31
00:02:22,182 --> 00:02:27,517
Let's figure out the equation of the 
tangent line to that graph at the point 

32
00:02:27,517 --> 00:02:31,214
(3,9). 
So to do that, we're going to use the 

33
00:02:31,214 --> 00:02:36,126
derivative and I know that the derivative 
of x^2 is 2x. 

34
00:02:36,126 --> 00:02:41,347
So, the slope of the tangent line at the 
point 3 is 2*3=6, 

35
00:02:41,347 --> 00:02:48,492
and that's the slope of the tangent line. 
I also know a point that the tangent line 

36
00:02:48,492 --> 00:02:52,100
passes through. 
It should be passing through the point 

37
00:02:52,100 --> 00:02:55,637
(3,9), right? The point (3,9) is on the 
tangent line. 

38
00:02:55,637 --> 00:03:00,371
So, if I know the slope of the tangent 
line and I know a point on the line, I 

39
00:03:00,371 --> 00:03:04,702
can write down an equation for the line 
using point slope form. 

40
00:03:04,702 --> 00:03:12,012
So, y minus the y coordinate on the line 
is the slope time x minus the x 

41
00:03:12,012 --> 00:03:19,834
coordinate on the line. 
So, this is an equation for this red 

42
00:03:19,834 --> 00:03:21,612
tangent line. 

43
00:03:21,612 --> 00:03:28,065
[MUSIC]