Answer
$3a^{2}+3ah+h^{2}$
Work Step by Step
$f(x)=x^{3}$
Find $f(a)$ by substituting $x$ with $a$:
$f(a)=a^{3}$
Find $f(a+h)$ by substituting $x$ with $a+h$:
$f(a+h)=(a+h)^{3}=a^{3}+3a^{2}h+3ah^{2}+h^{3}$
We have $f(a)$ and $f(a+h)$, we can substitute them into the expression $\dfrac{f(a+h)-f(a)}{h}$:
$\dfrac{f(a+h)-f(a)}{h}=\dfrac{a^{3}+3a^{2}h+3ah^{2}+h^{3}-a^{3}}{h}=...$
Simplify the numerator:
$...=\dfrac{3a^{2}h+3ah^{2}+h^{3}}{h}=...$
Take out common factor $h$ from the numerator and simplify the fraction:
$...=\dfrac{h(3a^{2}+3ah+h^{2})}{h}=3a^{2}+3ah+h^{2}$