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We've used the product rule to calculate some derivatives.

Â We've even seen a proof using limits, but there's still this nagging question,

Â why? For instance, why is there this + sign in the product rule? I mean, really,

Â with all those chiastic laws, the limit of a sum is the sum of the limits, limit

Â of products is the product of limits, you'd probably think the derivative of a

Â product is the product of the derivatives, I mean, you think that if

Â you differentiated a product, it'd just be the product of the derivatives.

Â No, that's not how products work.

Â What happens when you wiggle the terms in a product? We can explore this

Â numerically, so play around with this.

Â I've got a number a and another number b, and I'm multiplying them together to get

Â some new number, ab. initially, I've said a=2 and b=3, so

Â ab=6. But now I can wiggle the terms and see

Â how that affects the output. So what if I take a and move it from 2 to

Â 2.1? Well, that affects the output, the output is now 6.3.

Â Conversely what if I move that back down and I move b from 3 to 3.1? Well, that

Â makes the output from 6 to now 6.2. The deal here is that wiggling the input

Â affects the output by a magnitude that's related to the size of the other number,

Â right? When I went from 2 to 2.1, the output was affected by about three times

Â as much, the 3. When I moved the 3 from a 3 to a 3.1, the

Â output was affected by about two times as much and these affects add together.

Â What if I simultaneously move a from 2 to 2.1 and move b from 3 to 3.1, then the

Â output is 6.51, which is close to 6.5 which is what you guessed the answer

Â would be if you just add together these effects.

Â We can see the same thing geometrically. Geometrically, the product is really

Â measuring an area. So let me start with a rectangle of base

Â f(x) and height g(x). The product of f(x) and g(x) is then the

Â area of this rectangle. Now, I want to know how this area is

Â affected when I wiggle from x to say x+h. So lets suppose that I do that.

Â Let's suppose that I slightly change the size of the rectangle, so that now the

Â base isn't f(x) anymore, it's f(x+h) and the height isn't g(x) any more, it's

Â g(x+h). Now, how does the area change when the

Â input goes from x to x+h? Well, that's exactly just computing this

Â area and this L-shaped region here. I can do that approximately.

Â I actually know how much the base changes approximately, by using the derivative,

Â right? What's this length here approximately? Well, the derivative of f

Â at x times the input change is an approximation to how much the output

Â changes when I go from x to (x+h). So this distance is approximately f prime

Â of x times h. Same deal over here.

Â When the input goes from x to x+h, the output is changed by approximately the

Â derivative times the input change, so this length here is about g prime of x

Â times h. Now, I'm trying to compute the area of

Â this L-shaped region to figure out how the area, the product changes when I go

Â from x to x+h. Let me cut this L-shaped region up into

Â three pieces. This corner piece is pretty small, so I'm

Â going to end up disregarding that corner piece.

Â but let's just look at these two big pieces here.

Â This piece here is a rectangle and what's its area? Well, its base is f(x) and its

Â height is g prime of x times h. So the area of this piece, is f(x) times

Â g prime of x times h. What's the area of this rectangle over

Â here? Well, its base is f prime of x times h and its height is g(x), so the

Â area of this piece is f prime of x g of x times h Now, I want to know how did the

Â area change when I went from x to x+h? Well, that's pretty close to the, the sum

Â of these two rectangles. So the change in area is about f of x

Â times g prime of x times h plus f prime of x times g of x times h.

Â The derivative is the ratio of output change, which is about this, to input

Â change, which in this case is h. I went from x to x+h.

Â So now, I can cancel these h's, and what I'm left with is f of x times g

Â prime of x plus f prime x times g of x. That's the product rule.

Â That's the change in the area of this rectangle when I went from x to x+h

Â divided by how much I changed the input h.

Â The power rule isn't something that we just made up.

Â It's not some sort of sinister calculus plot designed to turn your mathematical

Â dreams into nightmares. This rule, the product rule, arises for

Â understandable reasons. If you wiggle one of the terms in a

Â product, the effect on the product has to do with the size of the other term.

Â You add together these two effects and then you have some idea as to how the

Â product changes based on how the terms change.

Â This is more than just a rule to memorize.

Â It's more that just a algorithm to apply. The product rule is telling you something

Â deep about how a product is effected when it's terms are changed.

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