Power Rule for Derivatives/Rational Index

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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
\(\ds \) \(=\) \(\ds n x^{n - 1}\)

$\blacksquare$


Sources