# Derivative of Function to Power of Function

## Theorem

Let $u \left({x}\right), v \left({x}\right)$ be real functions which are differentiable on $\R$.

Then:

$D_x \left({u^v}\right) = v u^{v-1} D_x \left({u}\right) + u^v \left({\ln u}\right) D_x \left({v}\right)$

## Proof

 $\displaystyle D_x \left({u^v}\right)$ $=$ $\displaystyle D_x \left({ \exp \left({v \ln u}\right) }\right)$ Power to Real Number $\displaystyle$ $=$ $\displaystyle \exp \left({v \ln u}\right) D_x \left({v \ln u}\right)$ Chain Rule for Derivatives and Derivative of Exponential Function $\displaystyle$ $=$ $\displaystyle \exp \left({v \ln u}\right) \left({\left({\ln u}\right) D_x \left({v}\right) + v D_x \left({\ln u}\right)}\right)$ Product Rule $\displaystyle$ $=$ $\displaystyle u^v \left({\left({\ln u}\right) D_x \left({v}\right) + \frac v u D_x \left({u}\right)}\right)$ Chain Rule for Derivatives $\displaystyle$ $=$ $\displaystyle v u^{v-1} D_x \left({u}\right) + u^v \left({\ln u}\right) D_x \left({v}\right)$ gathering terms

$\blacksquare$

## Also see

$D_x \left({x^n}\right) = n x^{n-1}$

$D_x \left({a^x}\right) = a^x \ln a$