# Evaluating a 'logarithmic' derivative

1. Sep 10, 2011

### ausdreamer

1. The problem statement, all variables and given/known data

I have to evaluate the following integral:

$\frac{\partial \log\rho (r)}{\partial \log r}$

for

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

where $\rho_0,\alpha,\beta$ are constants and $r$ is a random variable.

2. Relevant equations

-

3. The attempt at a solution

I've simplified the derivative to

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

but I'm stuck on where to go from here...I've almost finished my physics degree without encountering such a derivative :P

2. Sep 11, 2011

### diazona

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:
$$\frac{\partial}{\partial\log r} = \frac{\partial r}{\partial\log r}\frac{\partial}{\partial r} = \biggl(\frac{\partial\log r}{\partial r}\biggr)^{-1}\frac{\partial}{\partial r}$$
I'll let you take it from there

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