Find the derivative of x^1/x

by RoRi

November 13, 2015

Find the derivative of the function

$f(x) = x^{\frac{1}{x}}.$

We take the derivative using logarithmic differentiation. First, taking the logarithm of both sides,

$\log f(x) = \frac{1}{x} \log x = \frac{\log x}{x}.$

Then take the derivative of both sides,

$\begin{align*} &&\frac{f'(x)}{f(x)} &= \frac{1 - \log x}{x^2} \\ \implies && \frac{f'(x)}{f(x)} &= x^{-2}(1 - \log x)\\ \implies && f'(x) &= x^{-2 + \frac{1}{x}} (1 - \log x). \end{align*}$

Stumbling Robot

Find the derivative of x^1/x

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