Power Rule for Derivatives/Integer Index

Theorem
Let $n \in \Z$.

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
When $n \ge 0$ we use the result for Natural Number Index.

Now let $n \in \Z: n < 0$.

Then let $m = -n$ and so $m > 0$.

Thus $x^n = \dfrac 1 {x^m}$.