Now, many people probably know of the first three hyperoperations, such as addition, multiplication, and exponentiation. However, many don't know of the fourth, tetration, and even less then know about the fifth hyperoperation, pentation.
Much how like multiplication is repeated addition, and exponentiation is repeated multiplication. Tetration is repeated exponentiation.
E.g. $^2x$ = $x^x$ and $^3x$ = $x^{x^x}$
I was wondering if there was a general formula for the derivative of tetrations, turns out my friend and I found one for all n$\ge$2 where n$\in$$\mathbb{Z}$, this being.
$\frac{d}{dx}$[$^nx$] = $\displaystyle\sum_{k=1}^{n-2}$ $\left(_nx_{n-k}^{k-1}\right)\frac{1}{x}$ + $_nx$$_2^{n-2}$ + $_nx$$_2^{n-1}$
Take note of the notation im using here:
$_ax$$_z$ = $^ax$ * $^bx$* …$^zx$
$x^{k-1}$ = x * ln$^{k-1}(x)$
This will return the correct expression, but now I began to wonder if there was a formula for the derivative of pentated equation.
Pentation being: $_2x$=$^xx$
Which means x exponentiated, x many times. I have no clue where to even began differentiating this, the formula evidently wouldn't spit out a good answer and I've scoured the stack exchange and internet looking for an answer but all I've come across is a nice formula which I also found earlier with no prior knowledge of the post.
That being: $\frac{d}{dx}$[$^nx$] = $^nx$ $\left(\frac{^{n-1}x}{x}+ \frac{d}{dx}[^{n-1}x]ln(x)\right)$
Here's the link to the question: $n^{th}$ derivative of a tetration function
Despite this, it still doesn't help. I'm lost as to what to do so any help would be amazing. Alas, I'm just some random internet stranger not really knowing what he's talking about. So if I'm missing something painfully obvious feel free to point it out.
