Derivative of Power of Function/Proof 1

Theorem

Let $\map u x$ be a differentiable real function of $x$.

Let $n$ be a real number such that $n \ne -1$.

Then:

$\map {\dfrac \d {\d x} } {\map u x^n} = n \map u x^{n - 1} \map {\dfrac \d {\d x} } {\map u x}$

Proof

\(\ds \map {\frac \d {\d x} } {\map u x^n}\)

\(=\)

\(\ds \map {\frac \d {\d u} } {\map u x^n} \map {\frac \d {\d x} } {\map u x}\)

Chain Rule for Derivatives

\(\ds \)

\(=\)

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

Derivative of Hyperbolic Sine

$\blacksquare$