# 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'.

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