Norm Prokup

**Cornell University**

PhD. in Mathematics

Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and competes occasionally, despite his busy schedule.

Share

Use the product rule to find the derivative of the product of two functions--the first function times the derivative of the second, plus the second function times the derivative of the first. The **product rule** is related to the quotient rule, which gives the derivative of the quotient of two functions, and the chain rule, which gives the derivative of the composite of two functions.

I want to talk about how to take the derivative of a product. And we'll start by talking about how not to do it. Let me give you an example. I have two functions. F of X equals X to the 4th and G of X equals X to the 6th . So I can make a product out of the two functions. F times G and that'll turn out to be X to the 10th . Now what's the derivative of this product of the two functions?

Well you might expect the derivative to be, the derivative of the product to be the product of the derivatives. The derivative with respect to X of F of X times the derivative with respect to X of G of X. And that would give you... well the derivative of F of X would be the derivative of X to the 4th and that's 4X cubed.

And the derivative of G of X is the derivative of X to the sixth, which is 6X to the fifth. And multiplying that out, I get 24X to the eighth. Good. But, I can also do this derivative a different way. The derivative of F of X times G of X is the same as the derivative of X to the tenth. And that derivative is 10X to the ninth, a very different function from 24X to the eighth. What went wrong?

This is what went wrong. This statement's not true. So that's very important. When you take the derivative of a product, it does not equal the product of the derivatives. In general, the derivative of a product is not the derivative of F of X times the derivative of G of X. So, what is it? Well, what it is, is the product rule.

The derivative of F of X times G of X is the first function F of X times the derivative of the second function, G of X plus the second function G of X... (I'm going to go out of my box here)... times the derivative of F of X. The first times the derivative of the second, plus the second times the derivative of the first. That's the product rule.