$f(x) = \left\{ \begin{array}{ll} \ln(\sin(x))-\ln(x) & \mbox{if } 0<x<\pi \\ 0 \\ \ln(-\sin(x)) - \ln(-x) & \mbox{if } -\pi<x<0 \end{array} \right.$
The function is continuous, I've already checked that out. My problem is that I can not find whether the derivative exists at $x=0$.
Using the derivative by definition i get $$\lim_{h\to 0} \frac{\ln(\sin(h))-\ln(h)}{h}$$ and find that difficult to resolve. Thanks