When You Have Too Many Ex's Part 1

Calculus Level 4

What is the slope of the line tangent to the graph of

$\huge{y=x^{x^{x^{x^{x^{.^{.^{.}}}}}}}}$

at the point $$x=\sqrt{2}$$?

If your answer can be expressed as

$p\sqrt{q}\displaystyle \sum_{k=0}^{\infty}\ln^{k}(r)$

for positive, prime integers $$(p,q,r)$$, find $$p+q+r$$.

Note: the x's continue to infinity.

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