Derivative of Powers of Functions

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Theorem

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

Then:

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

$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle D_x \left({u^v}\right)$	$=$	$\displaystyle D_x \left({e^{v \ln u} }\right)$	$\displaystyle $	$\displaystyle $	$\displaystyle $	Powers of Real Numbers
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$=$	$\displaystyle e^{v \ln u} D_x \left({v \ln u}\right)$	$\displaystyle $	$\displaystyle $	$\displaystyle $	Chain Rule and Derivative of Exponential Function
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$=$	$\displaystyle e^{v \ln u} \left({\ln u D_x \left({v}\right) + v D_x \left({\ln u}\right)}\right)$	$\displaystyle $	$\displaystyle $	$\displaystyle $	Product Rule
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$=$	$\displaystyle u^v \left({\ln u D_x \left({v}\right) + \frac v u D_x \left({u}\right)}\right)$	$\displaystyle $	$\displaystyle $	$\displaystyle $	Chain Rule
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$=$	$\displaystyle v u^{v-1} D_x \left({u}\right) + u^v \ln u D_x \left({v}\right)$	$\displaystyle $	$\displaystyle $	$\displaystyle $	gathering terms

$\blacksquare$

When $u = x$ and $v = n$ where $n$ is constant, we get the Power Rule for Derivatives:

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

When $v = x$ and $u = a$ where $a$ is constant, we get the Derivative of Exponential Function:

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

\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle D_x \left({u^v}\right)\)	\(=\)	\(\displaystyle D_x \left({e^{v \ln u} }\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Powers of Real Numbers
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle e^{v \ln u} D_x \left({v \ln u}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Chain Rule and Derivative of Exponential Function
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle e^{v \ln u} \left({\ln u D_x \left({v}\right) + v D_x \left({\ln u}\right)}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Product Rule
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle u^v \left({\ln u D_x \left({v}\right) + \frac v u D_x \left({u}\right)}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Chain Rule
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle v u^{v-1} D_x \left({u}\right) + u^v \ln u D_x \left({v}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	gathering terms