Power Rule for Derivatives

Theorem
Let $n \in \R$.

Let $f: \R \to \R$ be the real function defined as $\map f x = x^n$.

Then:
 * $\map {f'} x = n x^{n - 1}$

everywhere that $\map f x = x^n$ is defined.

When $x = 0$ and $n = 0$, $\map {f'} x$ is undefined.

Proof
This can be done in sections.