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General expression for the derivative?

  1. Jun 25, 2011 #1
    I'm trying to find the general form for the nth derivative of
    f(x) = \frac{1}{x^m \log x}

    where m can be anything (set m = 1 for instance). For ease, you can take m to be integral.

    It sounds surprisingly simple, but the most I've been able to say is
    f^{(n)}(x) = (-1)^n x^{-(m+n)} \sum_{k=0}^n a_{k, n} [\log(x)]^{-k}

    where the coefficients satisfy
    a_{k,n} = [m + (n-1)] a_{k, n-1} + (k-1) a_{k-1, n-1}

    for 0 < k < n, and with [tex]a_{0, n} = (m + n - 1)!/(m-1)![/tex] and [tex]a_{n,n} = n![/tex]

    Unfortunately, I was hoping to get a general form for the coefficients. Does anyone know a trick?
  2. jcsd
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