Si $f (x) = 1/(x − 1)$,  calcular $f ′(x)$.

Solución:  \begin{align*}
f'(x)&=\lim_{h\rightarrow 0}\frac{f(x+h)-f(x)}{h}\\
&=\lim_{x\rightarrow 0}\frac{\frac{1}{(x+h)-1}-\frac{1}{x-1}}{h}\\
&=\lim_{h\rightarrow 0}\frac{\frac{x-1-(x+h-1)}{(x+h-1)(x-1)}}{h}\\
&=\lim_{h\rightarrow 0}\frac{\frac{-h}{(x+h-1)(x-1)}}{\frac{h}{1}}\\
&=\lim_{h\rightarrow 0}\frac{-1}{(x+h-1)(x-1)}\\
&=-\frac{1}{(x-1)^2}, \text{ por continuidad.}
\end{align*}