Derivative of a Composite Function

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.

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.