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}
$