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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