Finding the derivative of $x$ tetrated to the $x$

Differentiating the functions $$x^x$$, $$x^{x^x}$$ (or $${^2{x}}$$ and $${^3{x}}$$), etc., although somewhat tedious, is pretty straightforward. I've even seen in a couple of books (and even on a post on this forum) a general expression for the derivative of $${^n{x}}$$. However, I would like to find the derivative $${^x{x}}$$, or $$x$$ tetrated to the $$x$$. I seem to have absolutely no tools to even approach this problem, as I am not at all familiar with the maths associated with tetration.

• How do you define ${^y{x}}$ for $y \in\Bbb R$? – Martin R Nov 13 at 8:08
• Good point. It seems that there is no extension into the reals for tetration. – A. Lavie Nov 14 at 17:15

$$\frac{d}{dx}(^nx) = (^nx)\Bigl((lnx)\frac{d}{dx}(^{n-1}x)+\frac{^{n-1}x}{x}\Bigl)$$
• I am talking about the derivative for ${^x{x}}$, not $x^x$. – A. Lavie Nov 6 at 3:40