Power rule derivative in complex Problem:
Prove that if $f(z)= z^n$, then $f' (z)$ = $n z^{n-1} $ using the definition of the derivative.

Just write out: $$f(z+h) = \sum_{k=0}^n \binom{n}{k}z^{n-k}h^k = f(z) + h(nz^{n-1}) + h^2\cdot\sum\dots.$$
