1
00:00:00,012 --> 00:00:05,712
We've used the product rule to calculate 
some derivatives. 

2
00:00:05,712 --> 00:00:13,281
We've even seen a proof using limits, but 
there's still this nagging question, 

3
00:00:13,281 --> 00:00:20,709
why? For instance, why is there this + 
sign in the product rule? I mean, really, 

4
00:00:20,709 --> 00:00:25,280
with all those chiastic laws, the limit 
of a sum is the sum of the limits, limit 

5
00:00:25,280 --> 00:00:29,815
of products is the product of limits, 
you'd probably think the derivative of a 

6
00:00:29,815 --> 00:00:33,774
product is the product of the 
derivatives, I mean, you think that if 

7
00:00:33,774 --> 00:00:38,156
you differentiated a product, it'd just 
be the product of the derivatives. 

8
00:00:38,156 --> 00:00:40,148
No, 
that's not how products work. 

9
00:00:40,148 --> 00:00:44,742
What happens when you wiggle the terms in 
a product? We can explore this 

10
00:00:44,742 --> 00:00:47,912
numerically, 
so play around with this. 

11
00:00:47,912 --> 00:00:54,597
I've got a number a and another number b, 
and I'm multiplying them together to get 

12
00:00:54,597 --> 00:01:00,082
some new number, ab. 
initially, I've said a=2 and b=3, so 

13
00:01:00,082 --> 00:01:03,467
ab=6. 
But now I can wiggle the terms and see 

14
00:01:03,467 --> 00:01:09,449
how that affects the output. 
So what if I take a and move it from 2 to 

15
00:01:09,449 --> 00:01:16,191
2.1? Well, that affects the output, the 
output is now 6.3. 

16
00:01:16,191 --> 00:01:25,581
Conversely what if I move that back down 
and I move b from 3 to 3.1? Well, that 

17
00:01:25,581 --> 00:01:32,571
makes the output from 6 to now 6.2. 
The deal here is that wiggling the input 

18
00:01:32,571 --> 00:01:38,532
affects the output by a magnitude that's 
related to the size of the other number, 

19
00:01:38,532 --> 00:01:43,879
right? When I went from 2 to 2.1, the 
output was affected by about three times 

20
00:01:43,879 --> 00:01:47,840
as much, the 3. 
When I moved the 3 from a 3 to a 3.1, the 

21
00:01:47,840 --> 00:01:53,582
output was affected by about two times as 
much and these affects add together. 

22
00:01:53,582 --> 00:02:00,778
What if I simultaneously move a from 2 to 
2.1 and move b from 3 to 3.1, then the 

23
00:02:00,778 --> 00:02:07,525
output is 6.51, which is close to 6.5 
which is what you guessed the answer 

24
00:02:07,525 --> 00:02:12,162
would be if you just add together these 
effects. 

25
00:02:12,162 --> 00:02:19,572
We can see the same thing geometrically. 
Geometrically, the product is really 

26
00:02:19,572 --> 00:02:25,506
measuring an area. 
So let me start with a rectangle of base 

27
00:02:25,506 --> 00:02:31,129
f(x) and height g(x). 
The product of f(x) and g(x) is then the 

28
00:02:31,129 --> 00:02:36,168
area of this rectangle. 
Now, I want to know how this area is 

29
00:02:36,168 --> 00:02:42,410
affected when I wiggle from x to say x+h. 
So lets suppose that I do that. 

30
00:02:42,410 --> 00:02:49,121
Let's suppose that I slightly change the 
size of the rectangle, so that now the 

31
00:02:49,121 --> 00:02:56,404
base isn't f(x) anymore, it's f(x+h) and 
the height isn't g(x) any more, it's 

32
00:02:56,404 --> 00:02:59,415
g(x+h). 
Now, how does the area change when the 

33
00:02:59,415 --> 00:03:04,002
input goes from x to x+h? 
Well, that's exactly just computing this 

34
00:03:04,002 --> 00:03:08,486
area and this L-shaped region here. 
I can do that approximately. 

35
00:03:08,486 --> 00:03:14,243
I actually know how much the base changes 
approximately, by using the derivative, 

36
00:03:14,243 --> 00:03:19,480
right? What's this length here 
approximately? Well, the derivative of f 

37
00:03:19,480 --> 00:03:24,918
at x times the input change is an 
approximation to how much the output 

38
00:03:24,918 --> 00:03:31,128
changes when I go from x to (x+h). 
So this distance is approximately f prime 

39
00:03:31,128 --> 00:03:33,586
of x times h. 
Same deal over here. 

40
00:03:33,586 --> 00:03:40,084
When the input goes from x to x+h, the 
output is changed by approximately the 

41
00:03:40,084 --> 00:03:45,137
derivative times the input change, 
so this length here is about g prime of x 

42
00:03:45,137 --> 00:03:47,787
times h. 
Now, I'm trying to compute the area of 

43
00:03:47,787 --> 00:03:52,987
this L-shaped region to figure out how 
the area, the product changes when I go 

44
00:03:52,987 --> 00:03:56,587
from x to x+h. 
Let me cut this L-shaped region up into 

45
00:03:56,587 --> 00:04:00,237
three pieces. 
This corner piece is pretty small, so I'm 

46
00:04:00,237 --> 00:04:03,287
going to end up disregarding that corner 
piece. 

47
00:04:03,287 --> 00:04:06,692
but let's just look at these two big 
pieces here. 

48
00:04:06,692 --> 00:04:13,041
This piece here is a rectangle and what's 
its area? Well, its base is f(x) and its 

49
00:04:13,041 --> 00:04:17,802
height is g prime of x times h. 
So the area of this piece, is f(x) times 

50
00:04:17,802 --> 00:04:22,082
g prime of x times h. 
What's the area of this rectangle over 

51
00:04:22,082 --> 00:04:27,427
here? Well, its base is f prime of x 
times h and its height is g(x), so the 

52
00:04:27,427 --> 00:04:34,356
area of this piece is f prime of x g of x 
times h Now, I want to know how did the 

53
00:04:34,356 --> 00:04:41,802
area change when I went from x to x+h? 
Well, that's pretty close to the, the sum 

54
00:04:41,802 --> 00:04:47,626
of these two rectangles. 
So the change in area is about f of x 

55
00:04:47,626 --> 00:04:51,337
times g prime of x times h plus f prime 
of x times g of x times h. 

56
00:04:51,337 --> 00:04:59,177
The derivative is the ratio of output 
change, which is about this, to input 

57
00:04:59,177 --> 00:05:04,667
change, which in this case is h. 
I went from x to x+h. 

58
00:05:04,667 --> 00:05:11,177
So now, I can cancel these h's, 
and what I'm left with is f of x times g 

59
00:05:11,177 --> 00:05:16,677
prime of x plus f prime x times g of x. 
That's the product rule. 

60
00:05:16,677 --> 00:05:23,227
That's the change in the area of this 
rectangle when I went from x to x+h 

61
00:05:23,227 --> 00:05:26,932
divided by how much I changed the input 
h. 

62
00:05:26,932 --> 00:05:30,260
The power rule isn't something that we 
just made up. 

63
00:05:30,260 --> 00:05:35,342
It's not some sort of sinister calculus 
plot designed to turn your mathematical 

64
00:05:35,342 --> 00:05:39,501
dreams into nightmares. 
This rule, the product rule, arises for 

65
00:05:39,501 --> 00:05:43,280
understandable reasons. 
If you wiggle one of the terms in a 

66
00:05:43,280 --> 00:05:48,332
product, the effect on the product has to 
do with the size of the other term. 

67
00:05:48,332 --> 00:05:54,649
You add together these two effects and 
then you have some idea as to how the 

68
00:05:54,649 --> 00:05:58,657
product changes based on how the terms 
change. 

69
00:05:58,657 --> 00:06:02,205
This is more than just a rule to 
memorize. 

70
00:06:02,205 --> 00:06:09,298
It's more that just a algorithm to apply. 
The product rule is telling you something 

71
00:06:09,298 --> 00:06:14,861
deep about how a product is effected when 
it's terms are changed. 

72
00:06:14,861 --> 00:06:15,553
[MUSIC]