Question

If f(x) = (2x3 − 4)6, then what is f '(x)? (4 points)

Answer 1

Answer:

$f'(x)=36x^2(2x^3-4)^5$

Step-by-step explanation:

The Chain Rule for Derivatives

Given a function y=f(u) and u=g(x), the derivative dy/dx is computed by using the chain rule:

$y'=f'(u).u'(x)$

We have:

$f(x)=(2x^3-4)^6$

It can be written as:

$f(x)=u^6$

Where:

$u=2x^3-4$

Thus:

$f'(x)=(u^6)'(2x^3-4)'$

$f'(x)=6u^5(6x^2)$

Changing back u:

$f'(x)=6(2x^3-4)^5(6x^2)$

Operating:

$\mathbf{f'(x)=36x^2(2x^3-4)^5}$