What is the derivative of a sexp()

japplepie · Dec 22, 2014

d(x^^n)/dx = ?

moriheru · Dec 22, 2014

What is (x^^n)?

Mentallic · Dec 22, 2014

moriheru said: ↑

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:
[tex]x^{x^{.^{.^{.^x}}}}[/tex]

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

japplepie · Dec 23, 2014

Mentallic said: ↑

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

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 · Dec 23, 2014

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

japplepie · Dec 23, 2014

HallsofIvy said: ↑

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

Sorry, what I meant was positive integers.

DarthMatter · Dec 24, 2014

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 · Dec 24, 2014

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 · Dec 24, 2014

japplepie said: ↑

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.