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What is the derivative of a sexp()

  1. Dec 22, 2014 #1
    d(x^^n)/dx = ?
     
  2. jcsd
  3. Dec 22, 2014 #2
    What is (x^^n)?
     
  4. Dec 22, 2014 #3

    Mentallic

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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.
     
  5. Dec 23, 2014 #4
    Im only looking for the formula for integer values of x, will that make it simpler?
     
  6. Dec 23, 2014 #5

    HallsofIvy

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    That function is not even defined for integer values of x.
     
  7. Dec 23, 2014 #6
    Sorry, what I meant was positive integers.
     
  8. Dec 24, 2014 #7
    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)
     
  9. Dec 24, 2014 #8
    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: Dec 24, 2014
  10. Dec 24, 2014 #9
    It would be better if you type in latex.
     
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