Answer
$$f'\left( x \right) = 24x{\left( {3{x^2} - 2} \right)^3}$$
Work Step by Step
$$\eqalign{
& f\left( x \right) = {\left( {3{x^2} - 2} \right)^4} \cr
& {\text{differentiate both sides}} \cr
& f'\left( x \right) = \frac{d}{{dx}}\left[ {{{\left( {3{x^2} - 2} \right)}^4}} \right] \cr
& {\text{use the power rule with the chain rule }}\frac{d}{{dx}}\left[ {g{{\left( x \right)}^n}} \right] = ng{\left( x \right)^{n - 1}}\frac{d}{{dx}}\left[ {g'\left( x \right)} \right]{\text{ then}} \cr
& f'\left( x \right) = 4{\left( {3{x^2} - 2} \right)^3}\frac{d}{{dx}}\left[ {3{x^2} - 2} \right] \cr
& {\text{find derivative}} \cr
& f'\left( x \right) = 4{\left( {3{x^2} - 2} \right)^3}\left( {6x} \right) \cr
& {\text{simplify}} \cr
& f'\left( x \right) = 24x{\left( {3{x^2} - 2} \right)^3} \cr} $$
