I'm trying to find the general form for the nth derivative of(adsbygoogle = window.adsbygoogle || []).push({});

[tex]

f(x) = \frac{1}{x^m \log x}

[/tex]

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

[tex]

f^{(n)}(x) = (-1)^n x^{-(m+n)} \sum_{k=0}^n a_{k, n} [\log(x)]^{-k}

[/tex]

where the coefficients satisfy

[tex]

a_{k,n} = [m + (n-1)] a_{k, n-1} + (k-1) a_{k-1, n-1}

[/tex]

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?

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# General expression for the derivative?

Can you offer guidance or do you also need help?

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