For the recent week I've been diving myself into tetration out of pure curiosity, and after watching videos about how to find the derivative of $^2x$ and $^3x$, I decided that I wanted to look into what a derivative rule for it would be. When I searched online I couldn't find anything that was satisfying to me, so I put some work in and through a couple iterations of solutions, I think I have a closed form derivative rule I am personally satisfied with. What I found was:
$$\frac{d}{dx}(^nx)=\frac{^nx}{x} \sum_{a=0}^{\lceil n \rceil -1} \ln(x)^a \prod_{b=0}^a \frac{\ln(^{n-b}x)}{\ln(x)}.$$
Firstly, I've been wanting to get on this website for a while, so I thought I would bring it here. Secondly, I wanted to know if this looks accurate to anyone here, as through my testing with it, it seems to be. And lastly, I'm just a Calc BC student, exploring into these further realms of mathematics myself because it interests me, so I don't know, given that this is accurate, if there is a more concise or elegant way to put this derivative rule together, so is there a way I could?
A note about this is that I have it working for $n$ being any nonnegative real number, and for that to work, I am going by the recursive definition for tetration of:
$$^nx:=\begin{cases} 1 & \text{if } n = 0\cr x^{\big(^{(n-1)}x\big)} & \text{if } n > 0, \end{cases}$$
which extends it so that $n$ can be any nonnegative real number, such that $^{2.47}x$ would be considered $x^{x^{x^{0.47}}}$.
Update:
As I know that the recursive definition for tetration is not exactly the definition for tetration across the reals, and that that is still a researched topic, I wanted to share my derivative rule which I made before this one that works for $n$ as any nonnegative integer. Here is what I have on that:
$$\frac{d}{dx}(^nx)=\frac{^nx}{x}\sum_{a=0}^{n-1}\ln(x)^a\prod_{b=0}^a{^{n-b-1}x}.$$
For this, I ask the same questions as I asked above. Thank you for your time reading this.