What is the derivative of a sexp()

  • Thread starter japplepie
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  • #1
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Main Question or Discussion Point

d(x^^n)/dx = ?
 

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  • #2
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What is (x^^n)?
 
  • #3
Mentallic
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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.
 
  • #4
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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?
 
  • #5
HallsofIvy
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That function is not even defined for integer values of x.
 
  • #6
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That function is not even defined for integer values of x.
Sorry, what I meant was positive integers.
 
  • #7
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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. o0)
 
  • #8
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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?
 
Last edited:
  • #9
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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.
 
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