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.

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

$\ds \map D {x^n}$

$=$

$\ds \map D {\frac 1 {x^m} }$

$\ds $

$=$

$\ds \frac {x^m \cdot 0 - 1 \cdot m x^{m - 1} } {x^{2 m} }$

$\ds $

$=$

$\ds -m x^{-m - 1}$

$\ds $

$=$

$\ds n x^{n - 1}$

$\blacksquare$