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Evaluating a 'logarithmic' derivative

  1. Sep 10, 2011 #1
    1. The problem statement, all variables and given/known data

    I have to evaluate the following integral:

    [itex]\frac{\partial \log\rho (r)}{\partial \log r}[/itex]

    for

    [itex]\rho (r) = \rho_0 \Big(1+\big(\frac{r}{\alpha}\big)^2\Big)^\frac{-3 \beta}{2}[/itex]

    where [itex]\rho_0,\alpha,\beta[/itex] are constants and [itex]r[/itex] is a random variable.

    2. Relevant equations

    -

    3. The attempt at a solution

    I've simplified the derivative to

    [itex]\frac{\partial \log \rho (r)}{\partial \log r} = -\frac{3\beta}{2}\log \Bigg[\rho_0^\frac{-2}{3\beta} \bigg(1+\big(\frac{r}{A}\big)^2\bigg) \Bigg][/itex]

    but I'm stuck on where to go from here...I've almost finished my physics degree without encountering such a derivative :P
     
  2. jcsd
  3. Sep 11, 2011 #2

    diazona

    User Avatar
    Homework Helper

    This comes up a lot in certain fields of research-level physics, but you don't see it that much at undergrad level. Still, it's easy to evaluate using the chain rule:
    [tex]\frac{\partial}{\partial\log r} = \frac{\partial r}{\partial\log r}\frac{\partial}{\partial r} = \biggl(\frac{\partial\log r}{\partial r}\biggr)^{-1}\frac{\partial}{\partial r}[/tex]
    I'll let you take it from there :wink:

    EDIT: actually, your simplification doesn't seem quite right... in any case, I'd recommend starting by using the identity I mentioned. See if that gives you what you need, and if not, post your work in detail.
     
    Last edited: Sep 11, 2011
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