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

Question

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.

Answer 1

$$\frac{d}{dx}(^nx) = (^nx)\Bigl((lnx)\frac{d}{dx}(^{n-1}x)+\frac{^{n-1}x}{x}\Bigl)$$