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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