Cardinality of Infinite tetration

[bo198214]

What is |x[4]2|?
I was not aware that x[4]2 is a set.

Hmm. I thought it can be perceived as set by analogy to n^n which is a set of all permutations of n from n with order and repetitions.

Whatever, but \( n \) is a variable so you can regard \( 5^5 \) as a set in this sense but not \( n^n \) or \( x^x \) as it depends on \( n \) or \( x \) respectively.

Then next derivatives and their general form are easy to see:

\( x^{(x-n)}*x+x^{x}*(\ln(x))^n=x^{x-n+1}+x^{x}*(\ln(x))^n \)

Unfortunately its not that easy, e.g:
\( \frac{\partial^2 x^x}{(\partial x)^2} = {{x}^{x} {\left( \ln \left( x \right) + 1 \right)}^{2} } + {x}^{x - 1} \)