I happened to ponder about the differentiation of the following function: $$f(x)=x^{2x^{3x^{4x^{5x^{6x^{7x^{.{^{.^{.}}}}}}}}}}$$ Now, while I do know how to manipulate power towers to a certain extent, and know the general formula to differentiate $g(x)$ wrt $x$, where $$g(x)=f(x)^{f(x)^{f(x)^{f(x)^{f(x)^{f(x)^{f(x)^{.{^{.^{.}}}}}}}}}}$$ I'm still unable to figure out as to how I can adequately manipulate the function to differentiate it within its domain of convergence, if it exists.

Sub-Q: What is the domain over which the function converges? Does it have only a finite domain (meaning there's no necessity to discuss its derivative)?

General formula (making domanial assumptions for f(x) of course): $$g'(x)=\frac{g^2(x)f'(x)}{f(x)\left[1-g(x)\ln(f(x))\right]}$$

Note: This has been posted in stackexchange; a look at the (incomplete) answers and their respective comments' sections will give you an insight into what has been looked into so far - [http://math.stackexchange.com/questions/1592377/what-is-the-derivative-of-fx-x2x3x4x5x6x7x ]