My request is to prove that the function
$$
f(x) = \begin{cases}
\frac{1}{x}-
 \frac{1}{e^x-1} & if & x\neq 0 \\
\frac{1}{2} & if & x=0
\end{cases}
$$
is differentiable at $x=0$. I already show that it is continuous at $x=0$ by using the L'Hopital rule twice. But in differentiability I failed to achieve this despite using L'Hopital rule.

Any help is welcome