Power Rule for Derivatives/Fractional Index

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Theorem

Let $n \in \N_{>0}$.

Let $f: \R \to \R$ be the real function defined as $\map f x = x^{1 / n}$.


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 1

Let $n \in \N_{>0}$.

Thus, let $f \left({x}\right) = x^{1/n}$.

From the definition of the power to a rational number, or alternatively from the definition of the root of a number, $f \left({x}\right)$ is defined when $x \ge 0$.

(However, see the special case where $x = 0$.)

From Continuity of Root Function, $f \left({x}\right)$ is continuous over the open interval $\left({0 \,.\,.\, \infty}\right)$, but not at $x = 0$ where it is continuous only on the right.


Let $y > x$.

From Inequalities Concerning Roots:

$\forall n \in \N_{>0}: X Y^{1/n} \ \left|{x - y}\right| \le n X Y \ \left|{x^{1/n} - y^{1/n}}\right| \le Y X^{1/n} \ \left|{x - y}\right|$

where $x, y \in \left[{X \,.\,.\, Y}\right]$.

Setting $X = x$ and $Y = y$, this reduces (after algebra) to:

$\displaystyle \frac 1 {n y} y^{1/n} \le \frac {y^{1/n} - x^{1/n}} {y - x} \le \frac 1 {n x} x^{1/n}$

From the Squeeze Theorem, it follows that:

$\displaystyle \lim_{y \to x^+} \ \frac {y^{1/n} - x^{1/n}} {y - x} = \frac 1 {n x} x^{1/n} = \frac 1 n x^{\frac 1 n - 1}$


A similar argument shows that the left hand limit is the same.


Thus the result holds for $f \left({x}\right) = x^{1/n}$.

$\blacksquare$


Proof 2

Let $n \in \N_{>0}$.

Thus, let $f \left({x}\right) = y = x^{1/n}$.

Thus $f^{-1} \left({y}\right) = x = y^n$ from the definition of root.

So:

\(\displaystyle D x^{1/n}\) \(=\) \(\displaystyle \frac 1 {D y^n}\) Derivative of Inverse Function‎
\(\displaystyle \) \(=\) \(\displaystyle \frac 1 {n y^{n-1} }\) Power Rule for Derivatives: Integer Index
\(\displaystyle \) \(=\) \(\displaystyle \frac 1 {n \left({x^{1/n} }\right)^{n-1} }\)
\(\displaystyle \) \(=\) \(\displaystyle \frac 1 n x^{\frac 1 n - 1}\)

$\blacksquare$