Answer
$-\dfrac{2}{(a-1)(a+h-1)}$
Work Step by Step
$f(x)=\dfrac{2x}{x-1}$
Find $f(a)$ by substituting $x$ with $a$
$f(a)=\dfrac{2a}{a-1}$
Find $f(a+h)$ by substituting $x$ with $a+h$:
$f(a+h)=\dfrac{2(a+h)}{a+h-1}$
We have $f(a)$ and $f(a+h)$, substitute them in the expression $\dfrac{f(a+h)-f(a)}{h}$ and simplify:
$\dfrac{f(a+h)-f(a)}{h}=\dfrac{\dfrac{2(a+h)}{a+h-1}-\dfrac{2a}{a-1}}{h}=...$
Evaluate the substraction in the numerator and simplify:
$...=\dfrac{\dfrac{2(a+h)(a-1)-2a(a+h-1)}{(a-1)(a+h-1)}}{h}=...$
$...\dfrac{\dfrac{2(a^{2}-a+ah-h)-2a^{2}-2ah+2a}{(a-1)(a+h-1)}}{h}=...$
$...=\dfrac{\dfrac{2a^{2}-2a+2ah-2h-2a^{2}-2ah+2a}{(a-1)(a+h-1)}}{h}=...$
$...\dfrac{\dfrac{-2h}{(a-1)(a+h-1)}}{h}=-\dfrac{2}{(a-1)(a+h-1)}$
