Power Rule for Derivatives/Integer Index
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Theorem
Let $n \in \Z$.
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.
Proof
When $n \ge 0$ we use the result for Natural Number Index.
Now let $n \in \Z: n < 0$.
Then let $m = -n$ and so $m > 0$.
Thus $x^n = \dfrac 1 {x^m}$.
\(\ds \map D {x^n}\) | \(=\) | \(\ds \map D {\frac 1 {x^m} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac {x^m \cdot 0 - 1 \cdot m x^{m - 1} } {x^{2 m} }\) | Quotient Rule for Derivatives | |||||||||||
\(\ds \) | \(=\) | \(\ds -m x^{-m - 1}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds n x^{n - 1}\) |
$\blacksquare$
Sources
- 1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): $\S 10.11 \ (2)$