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