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:
 * $f^{\prime} \left({x}\right) = n x^{n-1}$

everywhere that $f \left({x}\right) = x^n$ is defined.

When $x = 0$ and $n = 0$, $f^{\prime} \left({x}\right)$ is undefined.

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

Then we have: