Single Variable Calculus, Chapter 1, 1.1, Section 1.1, Problem 24

Evaluate the difference quotient $\displaystyle \frac{f(a + h) - f(a)}{h}$ for the function $f(x) = x^3$


$
\begin{equation}
\begin{aligned}

\frac{f(a + h) - f(a)}{h} &= \frac{(a + h)^3 - a^3}{h}
&&( \text{Substitute} f(a+h) \text{ and } f(a) \text{ to the function} f(x),
\text{ then divide it by } h)\\
\\


&= \frac{a^3 + 3a^2h + 3ah^2 + h^3 -a^3}{h}
&&(\text{ Combine like terms})\\
\\

&= \frac{3a^2h + 3ah^2 + h^3}{h}
&&(\text{ Factor the numerator}) \\
\\

&= \frac{\cancel{h}(h^2 + 3a^2 + 3ah)}{\cancel{h}}
&& (\text{ Cancel out like terms})\\
\\
& = 3a^2 + h^2 + 3ah

\end{aligned}
\end{equation}
$