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Power tower differentials

  1. Sep 8, 2005 #1
    How Does One Find The Differentials Of Power Functions.
    Examples Like A[x]^b[x]^c[x]^d[x]........
    Where Those Are Functions Of X?
    In Cases Where These Functions Are Power Towers Of Another Variable,what Happens?
  2. jcsd
  3. Sep 8, 2005 #2


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    Gold Member

    First, you re-arrange the function:
    Suppose the function is
    Let [itex]w(x)=b(x)^{...}[/itex]
    The function rearanges to
    By the chain rule, the derivative is
    [tex]e^{w(x)\ln{u(x)}}\cdot \frac{d(w(x)\ln{u(x)})}{dx}[/tex]
    [tex]=u(x)^{w(x)}\cdot [\frac{w(x)u'(x)}{u(x)}+w'(x)\ln{u(x)}][/tex]
    To find w'(x), just apply this method again.
  4. Sep 8, 2005 #3
    This is basically the same thing as Euler just said, but explained a bit differently.

    You can use something called Logarithmic Differentiation. I'll show you an example:


    Now take the derivative of both sides and simplify:

  5. Sep 11, 2005 #4
    you mis understood my question,i meant a function raised to another and then another continously till infinity
  6. Sep 11, 2005 #5
    Then you let the exponent be another function, just like Euler said in his post. Then you can simplify a(x)b(x) using logarithmic differentiation.
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