The task is defined the following way:

"We have the function $\displaystyle f(x) = x^{x^{x^{\ldots}}}$

Find it's derivative!"

There is no more additional information about f(x) so, obviously we have to find its natural domain, and the set where it is differetiable.

I am pretty sure that such set, where f(x) is differetiable is the interval (0, 1] (not sure if it is the maximal).

I have found a formula, for the derivative, but it contains $\displaystyle x^{x^{x^{\ldots}}}$ terms (founding the inverse function leads to that formula). The problem is: can you find the finite formula for the derivative (does it exist???). Or maybe you can find the derivative written as an infinite series?

And in the end, one thing. Please, do not confuse the function with $\displaystyle \left(\left(\left(x^x\right)^x\right)^x\right)^{.. .}$ like I did in the first look. It is the function defined that way:

$\displaystyle

f_1 (x) = x

$

$\displaystyle

f_{n+1} (x) = x^{f_n (x)}

$

$\displaystyle

f (x) = \lim_{n \rightarrow \infty} f_n (x)

$