What is the derivative of a sexp()

[japplepie]

Main Question or Discussion Point

d(x^^n)/dx = ?

 

[moriheru]

What is (x^^n)?

 

[Mentallic]

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.

 

[japplepie]

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]

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

 

[japplepie]

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

Sorry, what I meant was positive integers.

 

[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.

 

[japplepie]

ok I got it

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

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

 

[Raghav Gupta]

ok I got it

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

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

It would be better if you type in latex.