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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