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Differetiate this?

  1. Dec 6, 2005 #1
    Differentiate this?

    I need some help to differentiate this function:
    y = x^x^x^x^...^x
    I am sure there's got to be some appropriate substitution for the x^ term. Any clues?
    Last edited: Dec 6, 2005
  2. jcsd
  3. Dec 6, 2005 #2


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    Homework Helper

    I'm assuming you meant the hyperpower function, which is the infinite power tower function. The x's go "all the way up".

    You should read that function as [tex]y = x^{x^{x^{x^....}}}[/tex], that is, evaluate from the top down.

    Then you can reexpress that as [tex]y = x^y[/tex]

    Take natural logs of both sides and differentiate implicitly.

    [tex]\ln{y} = y\ln{x}[/tex]

    [tex]\frac{y'}{y} = \frac{y}{x} + y'\ln{x}[/tex]

    Group the terms together.

    [tex]y'(\frac{1}{y} - \ln{x}) = \frac{y}{x}[/tex]

    And you can carry out further simplification yourself.
  4. Dec 6, 2005 #3


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    The given function is a so-called power tower.
    It looks that the given function had finitely many levels, though.

    You might try defining

    [tex]f:y\rightarrow y^{x}[/tex],

    and use f of f of ... of f and chain rule.
    Last edited: Dec 6, 2005
  5. Dec 9, 2005 #4
    Thanks Curious4131 and benorin!
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