Find the derivative of (log x)x / xlog x

Find the derivative of the function

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

To take this derivative we want to use logarithmic differentiation. To that end, take the logarithm of both sides of the equation for $f(x)$ ,

$\begin{align*} \log f(x) &= \log \left( \frac{(\log x)^x}{x^{\log x}} \right) \\ &= \log ((\log x)^x) - \log ( x^{\log x}) \\ &= x \log (\log x) - (\log x)^2. \end{align*}$

Now, we can differentiate both sides to obtain,

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

Stumbling Robot

Find the derivative of (log x)^x / x^{log x}

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