How would one find:
$$\frac{\mathrm d}{\mathrm dx}{}^xx?$$
where ${}^ba$ is defined by $${}^ba\stackrel{\mathrm{def}}{=}\underbrace{ a^{a^{\cdot^{\cdot^{\cdot^a}}}}}_{\text{$b$ times}}$$
Work so far
The interval that I am working in is $(0, \infty)$. It doesn't make much sense to consider negative numbers. Although there exists no extension to the reals for tetration I am going to assume that it exists. My theory is that it shouldn't change the algebra involved; (correct me if I am wrong).
Some visual analysis on the curve and you can see that it diverges to $+\infty$ extremely rapidly. This means that the derivative is going to have similar properties as well.
Let $f(x, y):={}^yx$ so we can rewrite our tetration as $f(x, x)$. Now using the definition of the total derivative: $D\;g(x, y)=\partial_xg(x, y)+\partial_yg(x, y)$. This should allow us to differentiate $f$.
$$D\;f(x,y)=\frac{\partial}{\partial x}{}^yx+\frac{\partial}{\partial y}{}^yx$$
Let's focus on the first partial derivative $\partial_x{}^yx$. This is just the case of differentiating a finite power tower as $y$ is treated constant.
Firstly looking at some examples do derive a general formula for $D\;\;{}^nx$:
$$
\begin{array}{c|c}
n & D\;\;{}^nx\\
\hline
0 & 0\\
1 & 1\\
2 & {}^2x(\log x + 1)\\
3 & {}^3x\times {}^2x\times x^{-1}(x\log x(\log x + 1)+1)
\end{array}
$$
It is easy to see that there is some pattern emerging however because of it's recursive nature I could not form a formula to describe it.
Edit
$$\dfrac{d}{dx}\left(e^{{}^nx \log(x)}\right)={}^{n+1}x\dfrac{d}{dx}\left({}^nx \log(x)\right)={}^{n+1}x\left(({}^nx)' \log(x)+\frac{{}^nx}{x}\right)$$
The recursive formula for the partial was pointed out in comments however an explicit formula would be more useful for this purpose.
The second partial derivative is interesting and relies on properties of tetration.
I was hoping for it to be similar to exponention such that $$D_y \;x^y=D_y\;e^{y\log x}=e^{y \log x}\log x\;D_y\;y=x^y\log x$$
However I am not sure of an '$e$ for tetration' but I hope it would be something like this:
$$D_y \;{}^yx=D_y\;{}^{y\;\text{slog} x}t={}^{y\;\text{slog} x}t\;\text{slog} x\;D_y\;y={}^yx\;\text{slog}\; x$$
Where $\text{slog}$ denotes the super logarithm (slogorithm), an inverse of tetration.
Edit
This may as well be a possible identity which can easily be applied to the above:
$$\text{slog}\;\left({}^yx\right)=\text{slog}^y\;(x)$$
I am unsure about using slogorithms and tetration in this way and I feel I might just be abusing notation.
Work on Tetration
I will update this section with more rigorous definitions and properties of tetration. I cannot prove all of them now.
For $x\in \Bbb R$ and $n \in \Bbb N$, $${}^nx:=\underbrace{ x^{x^{\cdot^{\cdot^{\cdot^x}}}}}_{\text{$n$ times}}\tag{1}$$
For $x\in\Bbb R$ and $a,b\in\Bbb N$?,
$${{}^b({}^ax)={}^{a^b}x\tag{$\not2$}}$$
Through simply algebra you can find that the above is not the case.
Update:
This is just differentiating the pentation function.
Best Answer
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!!!