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 $f(x) = x^3$


$\begin{equation} \begin{aligned} \frac{f(a + h) - f(a)}{h} &= \frac{(a + h)^3 - a^3}{h} &&( \text{Substitute} f(a+h) \text{ and } f(a) \text{ to the function} f(x), \text{ then divide it by } h)\\ \\ &= \frac{a^3 + 3a^2h + 3ah^2 + h^3 -a^3}{h} &&(\text{ Combine like terms})\\ \\ &= \frac{3a^2h + 3ah^2 + h^3}{h} &&(\text{ Factor the numerator}) \\ \\ &= \frac{\cancel{h}(h^2 + 3a^2 + 3ah)}{\cancel{h}} && (\text{ Cancel out like terms})\\ \\ & = 3a^2 + h^2 + 3ah \end{aligned} \end{equation}$