(c) By definition, $Log\,z:=\log|z|+i\arg z$ where $\,\arg z\,$ is defined only up to an integer multiple of $\,2\pi\,$, thus taking into account what for the corresponding real functions happens, we have: $e^{Log\,z}=e^{\log|z|+i\arg z}=e^{\log|z|}e^{i\arg z}=|z|e^{i\arg z}=z$ since the expression before the last to the right above is just the polar representation of the complex number $\,z\,$ .
From here, and knowing that $\,(e^z)'=e^z\,$, we get by the chain rule:
$e^{Log\,z}=z\Longrightarrow \left(e^{Log\,z}\right)'=(z)'\Longrightarrow (Log\,z)'e^{Log\,z}=1\Longrightarrow (Log\,z)'=\frac{1}{e^{Log\,z}}=\frac{1}{z}$
(d) Take $\,z=0\Longrightarrow \arg 0=\arg 1=2k\pi i\,\,,\,\,k\in\Bbb Z\,$ , so $Log\,(e^0)=Log\,1=\log|1|+i\arg 1=2k\pi i\,$ so the above value depends on the chosen branch for the logarithm and thus the equality is not necessarily true