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.