College Algebra, Chapter 3, 3.1, Section 3.1, Problem 40

Given the function $\displaystyle f(x) = \frac{2x}{x-1}$. Find $f(a)$, $f(a+h)$ and the difference quotient $\displaystyle \frac{f(a+h) - f(a)}{h}$ where $h \neq 0$

For $f(a)$
$\displaystyle f(a) = \frac{2a}{a-1}$ Replace $x$ by $a$

For $f(a+h)$
$\displaystyle f(a+h) = \frac{2(a+h)}{a+h-1}$

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

$
\begin{equation}
\begin{aligned}
\frac{f(a-h)-f(a)}{h} &= \frac{\frac{f(a+h)}{a+h-1} - \frac{2a}{a-1} }{h} && \text{Substitute } f(a+h) = \frac{2(a+h)}{a+h-1} \text{ and } f(a) = \frac{2a}{a-h}\\
\\
&= \frac{(2a+2h)(a-1)-2a(a+h-1)}{h(a-1)(a+h-1)} && \text{Get the LCD}\\
\\
&= \frac{2a^2 - 2a + 2ah - 2h - 2a^2 - 2ah + 2a}{h(a-1)(a+h-1)} && \text{Combine like terms}\\
\\
&= \frac{-2\cancel{h}}{\cancel{h}(a-1)(a+h-1)} && \text{Cancel out like terms}\\
\\
&= \frac{-2}{(a-1)(a+h-1)}
\end{aligned}
\end{equation}
$