Show that $e^{\operatorname{Log}(z)} = z$ and use this to evaluate the derivative of the function Log(z).
I have done the first part like this:
Letting $z = re^{i\theta}$,
$ \begin{align} e^{\operatorname{Log}(z)} & = e^{\operatorname{Log}(re^i\theta)} \\ \\ & = e^{\log r + i(\theta + 2k\pi)} \\ \\ & = e^{\log(r)}e^{i(\theta + 2k\pi)} \\ \\ & = r[\cos(\theta + 2k\pi) + i\sin(\theta + 2k\pi)] \\ \\ & = r[\cos(\theta\pi) + i\sin(\theta)] \\ \\ & = re^{i\theta} = z \end{align} $
But I can't see how I am supposed to make use of that to calculate the derivative of $\operatorname{Log}(z)$.