Study Guide

Derivative of a Product of Functions

Derivative of a Product of Functions

There's a convenient rule we can use to find the derivative of expressions like (3x)(sin x) or x2ex.

For the first expression, we know how to find the derivatives of 3x and sin x, but we don't yet have a way to find the derivative of their product. That's where the product rule comes in.

We use the product rule whenever there's a function that's a product of two other functions of x. If

f(x) = ({something with an x}) × ({something else with an x}),

then use the product rule to find f  '.

Unfortunately, the derivative of a product is not the product of the derivatives. It's not like the derivative of the sum of two functions.

Product Rule

The derivative of f(x) · g(x) is f ' (x) · g(x) + f ' (x) · g(x).

In words, if we have a product of two functions, then the derivative of the product is

"derivative of the first times the second plus the first times the derivative of the second."

The examples will give you plenty of practice actually using the product rule.