# What is the derivative of a sexp()

1. Dec 22, 2014

### japplepie

d(x^^n)/dx = ?

2. Dec 22, 2014

### moriheru

What is (x^^n)?

3. Dec 22, 2014

### Mentallic

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.

4. Dec 23, 2014

### japplepie

Im only looking for the formula for integer values of x, will that make it simpler?

5. Dec 23, 2014

### HallsofIvy

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

6. Dec 23, 2014

### japplepie

Sorry, what I meant was positive integers.

7. Dec 24, 2014

### DarthMatter

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.

8. Dec 24, 2014

### japplepie

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: Dec 24, 2014
9. Dec 24, 2014

### Raghav Gupta

It would be better if you type in latex.