How to find the derivative function of g(x) = 3/(x-2)? f(x)= f(x+h)-f(x) / h

I guess you want the derivative of $g \left(x\right)$ using your definition of derivative (for $f \left(x\right)$ in your question).
${\lim}_{h \to 0} \frac{f \left(x + h\right) - f \left(x\right)}{h} = f ' \left(x\right)$