# Differentiating an Infinite Tetration

Calculus Level 4

Let $f: [e^{-e}, e^{1/e}] \mapsto [e^{-1},e]$ be defined as $f(x) = x^{x^{x^{.^{.^{.}}}}}$.

If $f'(\sqrt{2}) = \dfrac{a\sqrt{b}}{c-\ln{d}}$ for positive integers $a$, $b$, $c$ and $d$ where $b$ is square-free and $d$ is a prime number, find $a+b+c+d$.

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