Single Variable Calculus, Chapter 1, 1.1, Section 1.1, Problem 25

Evaluate the difference quotient $\displaystyle \frac{f(x) - f(a)}{x - a}$ for the function $\displaystyle f(x) = \frac{1}{x}$


$\begin{equation} \begin{aligned} \frac{f(x) - f(a)}{x - a} &= \frac{\frac{1}{x} - \frac{1}{a}}{x - a} && ( \text{Substitute $f(x)$ and $f(a)$ to the function $f(x)$, then divide it by $(x-a)$ and get the LCD})\\ \\ &= \frac{a - x}{ax(x-a)} &&( \text{ Simplify the equation})\\ \\ &= \frac{-1 \cancel{(x - a)}}{ax \cancel{(x - a)}} &&( \text{ Factor -1 in the numerator, and then cancel out like terms})\\ \\ & = \frac{-1}{ax} \end{aligned} \end{equation}$