# How do you find the derivative of f(x) = (1/x)^x?

##### 1 Answer
Jun 1, 2016

$\frac{\mathrm{dy}}{\mathrm{dx}} = - {x}^{- x} \left(1 + {\log}_{e} x\right)$

#### Explanation:

$y = {x}^{- x} \to {\log}_{e} y = - x {\log}_{e} x$
$\frac{\mathrm{dy}}{y} = - \mathrm{dx} {\log}_{e} x - \frac{x}{x} \mathrm{dx} = - \left(1 + {\log}_{e} x\right) \mathrm{dx}$
then
$\frac{\mathrm{dy}}{\mathrm{dx}} = - y \left(1 + {\log}_{e} x\right) = - {x}^{- x} \left(1 + {\log}_{e} x\right)$