What is the derivative of ${}^xx$

It seems to me that before much more progress can be made in the calculus of ${}^xy$, more fundamental questions have to be answereed, such as, how to define ${}^xy$ for rational $x$? It's clear how the OP's definition works if $x$ is a non-negative integer; but how do we define ${}^xy$ if, say, $x = 7/2$? What then is "one-half" of an occurrance of $x$ in the exponential "tower" which is supposed to be ${}^xy$?

I am reminded here of the way $x^y$ is extended from integers through the reals, by starting with a careful, consistent and believable definition of $(p / q)^{(r / s)}$ for integral $p, q, r, s$; once we have that, a simple, consistent and believable continuity argument allows us to accept a definition of $x^y$ for real $x, y > 0$. We know what $(p / q)^r = (p^r / q^r)$ means; we know what it means for a positive real $z$ to satisfy $z^s = (p / q)^r$, so we can get a handle on $(p / q)^{(r / s)}$ from which, by continuity, we can generalize to $x^y$. I think an analogous method is needed here, but I don't know what it is. But I think my question of the preceding paragraph might be worth considering early on in this game.

Of course, perhaps there is a (reasonably) simple, consistent and believable argument to contruct ${}^xy$ using $\exp()$, $\log()$, etc., or some sort of differential or similar equation ${}^xy$ must satisfy, or perhaps one could learn something from the $\Gamma$ function and factorials here which would bypass, at least temporarily, the need to address how ${}^{(p / q)}(r / s)$ is supposed to work, but sooner or later the question will have to be faced, I'll warrant.

This is an interesting, though speculative, arena and I am glad to have participated. But until I can answer my own questions to my better satisfaction, I will refrain from further remarks, except to bid those who are ready to climb such unknown heights, "Excelsior!

Hope this helps, at least with the spirit of the adventure if not with the direction. Happy New Year,

and as always,

Fiat Lux!!!

Your "$e$ for tetration" cannot exist. Indeed let $T(x,y)={}^yx$ be the extended tetration and let $t$ be the analogue to $e$ for tetration, in the sense that $$\partial_yT(x,y)=\partial_yT(t,y\operatorname{slog}x) = T(t,y\operatorname{slog}x)\operatorname{slog}x$$ Here $\operatorname{slog}x$ is the (unique, if $T$ is strictly increasing in its first argument) positive real number such that $x=T(x,1)=T(t,\operatorname{slog}x)$. Then $\operatorname{slog}t=1$ and thus by the above we have $$\partial_yT(t,y)=T(t,y)$$ which implies $T(t,y)=T(t,0)e^y=e^y$, which is obviously a contradiction.