Find the derivative of xxx - Stumbling Robot

Find the derivative of the following function:

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

Here we use logarithmic differentiation. Taking the logarithm of both sides we have

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

Now differentiating both sides gives us,

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

Stumbling Robot

Find the derivative of x^{x^x}

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