# What is the derivative of a sexp()

## Main Question or Discussion Point

d(x^^n)/dx = ?

What is (x^^n)?

Mentallic
Homework Helper
What is (x^^n)?
It's tetration which represents a double up arrow in Knuth's up arrow notation. Basically a power tower of x's n high:
$$x^{x^{.^{.^{.^x}}}}$$

And I don't see any simple solution to this problem.

It's tetration which represents a double up arrow in Knuth's up arrow notation. Basically a power tower of x's n high:
$$x^{x^{.^{.^{.^x}}}}$$

And I don't see any simple solution to this problem.
Im only looking for the formula for integer values of x, will that make it simpler?

HallsofIvy
Homework Helper
That function is not even defined for integer values of x.

That function is not even defined for integer values of x.
Sorry, what I meant was positive integers.

You can rewrite $x^x = \exp(\ln(x)\cdot x)$. Similiar $x^{x^x}=\exp(\exp(\ln(x)\cdot x)\cdot \ln(x))$. But you'll have to derivate yourself, I'm too lazy right now.

ok I got it

d(x ^^ n) / dx = x ^^ n * d(x ^^ ( n -1) * ln x ) / dx

lim i→ ∞ { di(x ^^ n) / dxi } converge?

Last edited:
ok I got it

d(x ^^ n) / dx = x ^^ n * d(x ^^ ( n -1) * ln x ) / dx