Power Rule for Derivatives/Rational Index

Theorem

Let $n \in \Q$.

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.

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 $

$=$

$\ds \frac 1 q \paren {x^p}^{1 / q} x^{-p} p x^{p - 1}$

$\ds $

$=$

$\ds \frac p q x^{\frac p q - 1}$

after some algebra

$\ds $

$=$

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

$\blacksquare$

1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): $\S 10.15 \ (2)$