Find the derivative of xlog x

Find the derivative of the function

$f(x) = x^{\log x}.$

To find this derivative we want to use logarithmic differentiation. Take the logarithm of both sides,

$log f(x) = \log (x^{\log x}) = (\log x)\cdot (\log x) = (\log x)^2.$

Then differentiating both sides of the equation (and using the chain rule for the derivative on the right hand side),

$\begin{align*} && \big(\log (f(x))\big)' &= \big( (\log x)^2 \big)' \\[9pt] \implies && \frac{f'(x)}{f(x)} &= 2 (\log x) (\log x)' \\[9pt] \implies && \frac{f'(x)}{f(x)} &= 2 (\log x) \left( \frac{1}{x} \right) \\[9pt] \implies && \frac{f'(x)}{f(x)} &= 2x^{-1} \log x \\[9pt] \implies && f'(x) &= \big( 2x^{-1} \log x \big) \cdot f(x). \end{align*}$

Substituting back in our original function $f(x) = x^{\log x}$ we obtain

$\begin{align*} f'(x) &= \big( 2x^{-1} \log x \big) \cdot (x^{\log x}) \\[9pt] &= 2 x^{-1 + \log x} \log x. \end{align*}$

Stumbling Robot

Find the derivative of x^{log x}

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