Power Rule for Derivatives/Fractional Index/Proof 2

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

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

Thus, let $\map f x = y = x^{1/n}$.

Thus $\map {f^{-1} } y = x = y^n$ from the definition of root.

So:

\(\ds D x^{1/n}\)

\(=\)

\(\ds \frac 1 {D y^n}\)

Derivative of Inverse Function‎

\(\ds \)

\(=\)

\(\ds \frac 1 {n y^{n - 1} }\)

Power Rule for Derivatives: Integer Index

\(\ds \)

\(=\)

\(\ds \frac 1 {n \paren {x^{1/n} }^{n - 1} }\)

\(\ds \)

\(=\)

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

$\blacksquare$