How do you differentiate f(x)=ln(sin(e^{x}))?

You can use the Chain Rule where you first derive $\ln$ as it is then multiply by the derivative of $\sin$ as it is and finally multiply by the derivative of $e$:
$f ' \left(x\right) = \frac{1}{\sin \left({e}^{x}\right)} \cdot \cos \left({e}^{x}\right) \cdot {e}^{x} = {e}^{x} \cdot \cot \left({e}^{x}\right)$