# Power Rule for Derivatives/Rational Index

## Theorem

Let $n \in \Q$.

Let $f: \R \to \R$ be the real function defined as $f \left({x}\right) = x^n$.

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

Let $n \in \Q$, such that $n = \dfrac p q$ where $p, q \in \Z, q \ne 0$.

Then we have:

 $\ds \map D {x^n}$ $=$ $\ds \map D {x^{p/q} }$ $\ds$ $=$ $\ds \map D {\paren {x^p}^{1/q} }$ $\ds$ $=$ $\ds \frac 1 q \paren {x^p}^{1/q} x^{-p} p x^{p-1}$ Chain Rule for Derivatives $\ds$ $=$ $\ds \frac p q x^{\frac p q - 1}$ after some algebra $\ds$ $=$ $\ds n x^{n - 1}$

$\blacksquare$