Derivative of a Composite Function

How to find the derivative of a composite function g(f(x)): formula (chain rule), 1 example, and its solution.

Formula

Derivative of g(f(x)): Chain Rule

The derivative of a composite function g(f(x)) is
g'(f(x))⋅f'(x).

First differentiate the outer function g.
Then differentiate the inner function f.

Example

Example

Solution

See the given function.
(2x2 - 1) is in (2x2 - 1)8.

So this is a composite function.

So use the chain rule.

Write y' =.

See (2x2 - 1) as a whole.
And write the derivative of (2x2 - 1)8:
8(2x2 - 1)7.

Derivative of a Polynomial

Write the derivative of the inner function (2x2 - 1):
2⋅2x1 - 0.

So y' = 8(2x2 - 1)7⋅(2⋅2x1 - 0).

(2⋅2x1 - 0) = 4x

8⋅4x = 32x

So y' = 32x(2x2 - 1)7 is the derivative of the given function.