Answer
$f'\left( x \right) = 4{x^3} + 9{x^2}$
Work Step by Step
$$\eqalign{
& f\left( x \right) = {x^3}\left( {x + 3} \right) \cr
& {\text{Rewrite the function using the distributive property}} \cr
& f\left( x \right) = {x^4} + 3{x^3} \cr
& {\text{Differentiate the function}} \cr
& f'\left( x \right) = \frac{d}{{dx}}\left[ {{x^4} + 3{x^3}} \right] \cr
& {\text{Use the sum rule for differentiation }} \cr
& f'\left( x \right) = \frac{d}{{dx}}\left[ {{x^4}} \right] + \frac{d}{{dx}}\left[ {3{x^3}} \right] \cr
& {\text{Use the constant multiple rule}} \cr
& f'\left( x \right) = \frac{d}{{dx}}\left[ {{x^4}} \right] + 3\frac{d}{{dx}}\left[ {{x^3}} \right] \cr
& {\text{Apply the power rule: }}\frac{d}{{dx}}\left[ {{x^n}} \right] = n{x^{n - 1}}{\text{ }} \cr
& f'\left( x \right) = 4{x^3} + 3\left( {3{x^2}} \right) \cr
& f'\left( x \right) = 4{x^3} + 9{x^2} \cr} $$
