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.

Suppose $f(z)=z^n.$ Then for all $a,$ we have
$$\begin{align}f'(a) &= \lim_{z\to a}\frac{f(z)-f(a)}{z-a}\\ &= \lim_{z\to a}\frac{z^n-a^n}{z-a}\\ &= \lim_{z\to a}\frac{(z-a)(z^{n-1}+az^{n-2}+a^2z^{n-3}+\cdots+a^{n-2}z+a^{n-1})}{z-a}\\ &= \lim_{z\to a}(z^{n-1}+az^{n-2}+a^2z^{n-3}+\cdots+a^{n-2}z+a^{n-1})\\ &= \underbrace{a^{n-1}+a^{n-1}+\cdots+a^{n-1}}_{n\text{ times}}\\ &= na^{n-1}.\end{align}$$
