We've used the product rule to calculate 
some derivatives. 
We've even seen a proof using limits, but 
there's still this nagging question, 
why? For instance, why is there this + 
sign in the product rule? I mean, really, 
with all those chiastic laws, the limit 
of a sum is the sum of the limits, limit 
of products is the product of limits, 
you'd probably think the derivative of a 
product is the product of the 
derivatives, I mean, you think that if 
you differentiated a product, it'd just 
be the product of the derivatives. 
No, 
that's not how products work. 
What happens when you wiggle the terms in 
a product? We can explore this 
numerically, 
so play around with this. 
I've got a number a and another number b, 
and I'm multiplying them together to get 
some new number, ab. 
initially, I've said a=2 and b=3, so 
ab=6. 
But now I can wiggle the terms and see 
how that affects the output. 
So what if I take a and move it from 2 to 
2.1? Well, that affects the output, the 
output is now 6.3. 
Conversely what if I move that back down 
and I move b from 3 to 3.1? Well, that 
makes the output from 6 to now 6.2. 
The deal here is that wiggling the input 
affects the output by a magnitude that's 
related to the size of the other number, 
right? When I went from 2 to 2.1, the 
output was affected by about three times 
as much, the 3. 
When I moved the 3 from a 3 to a 3.1, the 
output was affected by about two times as 
much and these affects add together. 
What if I simultaneously move a from 2 to 
2.1 and move b from 3 to 3.1, then the 
output is 6.51, which is close to 6.5 
which is what you guessed the answer 
would be if you just add together these 
effects. 
We can see the same thing geometrically. 
Geometrically, the product is really 
measuring an area. 
So let me start with a rectangle of base 
f(x) and height g(x). 
The product of f(x) and g(x) is then the 
area of this rectangle. 
Now, I want to know how this area is 
affected when I wiggle from x to say x+h. 
So lets suppose that I do that. 
Let's suppose that I slightly change the 
size of the rectangle, so that now the 
base isn't f(x) anymore, it's f(x+h) and 
the height isn't g(x) any more, it's 
g(x+h). 
Now, how does the area change when the 
input goes from x to x+h? 
Well, that's exactly just computing this 
area and this L-shaped region here. 
I can do that approximately. 
I actually know how much the base changes 
approximately, by using the derivative, 
right? What's this length here 
approximately? Well, the derivative of f 
at x times the input change is an 
approximation to how much the output 
changes when I go from x to (x+h). 
So this distance is approximately f prime 
of x times h. 
Same deal over here. 
When the input goes from x to x+h, the 
output is changed by approximately the 
derivative times the input change, 
so this length here is about g prime of x 
times h. 
Now, I'm trying to compute the area of 
this L-shaped region to figure out how 
the area, the product changes when I go 
from x to x+h. 
Let me cut this L-shaped region up into 
three pieces. 
This corner piece is pretty small, so I'm 
going to end up disregarding that corner 
piece. 
but let's just look at these two big 
pieces here. 
This piece here is a rectangle and what's 
its area? Well, its base is f(x) and its 
height is g prime of x times h. 
So the area of this piece, is f(x) times 
g prime of x times h. 
What's the area of this rectangle over 
here? Well, its base is f prime of x 
times h and its height is g(x), so the 
area of this piece is f prime of x g of x 
times h Now, I want to know how did the 
area change when I went from x to x+h? 
Well, that's pretty close to the, the sum 
of these two rectangles. 
So the change in area is about f of x 
times g prime of x times h plus f prime 
of x times g of x times h. 
The derivative is the ratio of output 
change, which is about this, to input 
change, which in this case is h. 
I went from x to x+h. 
So now, I can cancel these h's, 
and what I'm left with is f of x times g 
prime of x plus f prime x times g of x. 
That's the product rule. 
That's the change in the area of this 
rectangle when I went from x to x+h 
divided by how much I changed the input 
h. 
The power rule isn't something that we 
just made up. 
It's not some sort of sinister calculus 
plot designed to turn your mathematical 
dreams into nightmares. 
This rule, the product rule, arises for 
understandable reasons. 
If you wiggle one of the terms in a 
product, the effect on the product has to 
do with the size of the other term. 
You add together these two effects and 
then you have some idea as to how the 
product changes based on how the terms 
change. 
This is more than just a rule to 
memorize. 
It's more that just a algorithm to apply. 
The product rule is telling you something 
deep about how a product is effected when 
it's terms are changed. 
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