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Find the derivative of (log x)x

Find the derivative of the following function:

    \[ f(x) = (\log x )^x. \]


We want to use logarithmic differentiation, so first take the logarithm of both sides,

    \[ \log f(x) = x log (log x). \]

Then differentiating both sides we have

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

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