# The chain rule

The combined function can be differentiated using the following rule This is called a chain rule. The derivative of the combined function is thus the derivative of the outer function u' at s(x) times the derivative of the inner function s'.

Example 1

Differentiate the function Identify the function as a combined function. The outer and inner functions are Derivatives The derivative of function f with the chain rule Example 2

Differentiate the function The outer and inner functions are The derivative of function g with the chain rule Example 3

Find the zeros of the derivative A function consists of the product of two functions, one of which is a combined function. The product derivation formula is used We use the chain rule on the latter function Both terms of the derivative have a common factor We take the common factor Now we can use the zero-product property to find the zeros And zeros from these The zeros of the derivative function of the function f are x = -2, x = -4 /5 and x = 0

In the figure, green is the graph of the function f and green is the graph of the derivative function f'. Turn on the subtitles if needed