College Algebra, Chapter 3, 3.1, Section 3.1, Problem 42

Given the function $\displaystyle f(x) = x^3$. Find $f(a)$, $f(a+h)$ and the difference quotient $\displaystyle \frac{f(a+h) - f(a)}{h}$ where $h \neq 0$

For $f(a)$
$\displaystyle f(a) = a^3$ Replace $x$ by $a$

For $f(a+h)$

$\begin{equation} \begin{aligned} f(a+h) &= (a+h)^3 && \text{Replace } x \text{ by } (a + h)\\ \\ &= a^3 + 3a^2h + 3ah^2 + h^3 \end{aligned} \end{equation}$


For $\displaystyle \frac{f(a+h)-f(a)}{h}$

$\begin{equation} \begin{aligned} \frac{f(a-h)-f(a)}{h} &= \frac{a^3 + 3a^2h + 3ah^2 + h^3 - a^3}{h} && \text{Substitute } f(a+h) = a^3 + 3a^h + 3ah^2 + h^3 \text{ and } f(a) = a^3\\ \\ &= \frac{3a^2h + 3ah^2 + h^3}{h} && \text{Simplify}\\ \\ &= \frac{\cancel{h}\left( 3a^2 + 3ah + h^2 \right)}{\cancel{h}} && \text{Cancel out like terms}\\ \\ &= 3a^2 + 3ah + h^2 \end{aligned} \end{equation}$