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