Differentiating an Infinite Tetration

Calculus Level 4

Let f:[ee,e1/e][e1,e]f: [e^{-e}, e^{1/e}] \mapsto [e^{-1},e] be defined as f(x)=xxx...f(x) = x^{x^{x^{.^{.^{.}}}}}.

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

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