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