# Derivative of Function to Power of Function

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## Theorem

Let $\map u x, \map v x$ be real functions which are differentiable on $\R$.

Then:

$\map {\dfrac \d {\d x} } {u^v} = v u^{v - 1} \map {\dfrac \d {\d x} } u + u^v \paren {\ln u} \map {\dfrac \d {\d x} } v$

## Proof

 $\ds \map {\dfrac \d {\d x} } {u^v}$ $=$ $\ds \map {\dfrac \d {\d x} } {\map \exp {v \ln u} }$ Definition of Power to Real Number $\ds$ $=$ $\ds \map \exp {v \ln u} \map {\dfrac \d {\d x} } {v \ln u}$ Chain Rule for Derivatives and Derivative of Exponential Function $\ds$ $=$ $\ds \map \exp {v \ln u} \paren {\paren {\ln u} \map {\dfrac \d {\d x} } v + v \map {\dfrac \d {\d x} } {\ln u} }$ Product Rule for Derivatives $\ds$ $=$ $\ds u^v \paren {\paren {\ln u} \map {\dfrac \d {\d x} } v + \frac v u \map {\dfrac \d {\d x} } u}$ Chain Rule for Derivatives $\ds$ $=$ $\ds v u^{v - 1} \map {\dfrac \d {\d x} } u + u^v \paren {\ln u} \map {\dfrac \d {\d x} } v$ gathering terms

$\blacksquare$

## Also see

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

$\map {\dfrac \d {\d x} } {a^x} = a^x \ln a$