# What is the derivative of x^(cosx)?

Jun 3, 2016

$\frac{\mathrm{dy}}{\mathrm{dx}} = {x}^{\cos \left(x\right)} \left(\cos \frac{x}{x} - \sin \left(x\right) {\log}_{e} x\right)$

#### Explanation:

$y = {x}^{\cos \left(x\right)} \equiv {\log}_{e} y = \cos \left(x\right) {\log}_{e} x$
Deriving the log transformed equation we have
$\frac{\mathrm{dy}}{y} = - \sin \left(x\right) {\log}_{e} x \mathrm{dx} + \cos \frac{x}{x} \mathrm{dx}$
grouping
$\frac{\mathrm{dy}}{\mathrm{dx}} = y \left(\cos \frac{x}{x} - \sin \left(x\right) {\log}_{e} x\right)$
and finally
$\frac{\mathrm{dy}}{\mathrm{dx}} = {x}^{\cos \left(x\right)} \left(\cos \frac{x}{x} - \sin \left(x\right) {\log}_{e} x\right)$