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

Find the derivative of the following function:

    \[ f(x) = \log (\log (\log x)). \]


Here we apply the chain rule (carefully),

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

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