U.S.S. Alabama 2.4 Chain Rule Mobile, Alabama Greg Kelly, Hanford High School, Richland, Washington Photo by Vickie Kelly, 2002

Objectives Find the derivative of a composite function using the Chain Rule. Find the derivative of a function using the General Power Rule. Simplify the derivative of a function using algebra.

We now have a pretty good list of “shortcuts” to find derivatives of simple functions. Of course, many of the functions that we will encounter are not so simple. What is needed is a way to combine derivative rules to evaluate more complicated functions.

Consider a simple composite function:

and another:

and one more: This pattern is called the chain rule.

Chain Rule: If is the composite of and , then:

Chain Rule: If is the composite of and , then: Proof:

Example: Find the derivative

General Power Rule: Derivative formulas include the chain rule!

Every derivative problem could be thought of as a chain-rule problem: The most common mistake on this test is to forget to use the chain rule. Every derivative problem could be thought of as a chain-rule problem: The derivative of x is one. derivative of outside function derivative of inside function

Example: Find the derivative

Example: Find the derivative

Example: Find the derivative

Example: Find the derivative

Example: Find the derivative

Example: Find the derivative

Homework 2.4 (page 137) #7-39 odd