# Natural log derivatives

1. Aug 4, 2012

### Cinitiator

1. The problem statement, all variables and given/known data
What does it mean when the derivative of a function f(x) is in the form:

d ln f(x) / d ln x
?

Is it the logarithmic scale derivative, or something?

2. Relevant equations
d ln f(x) / d ln x

3. The attempt at a solution
Googling.

2. Aug 4, 2012

### SammyS

Staff Emeritus
By the chain rule:

$\displaystyle \frac{d\ \ln(f(x))}{dx}$
$\displaystyle = \frac{d\ \ln(f(x))}{d\ \ln(x)}\cdot\frac{d\ \ln(x)}{dx}$

$\displaystyle = \frac{d\ \ln(f(x))}{d\ \ln(x)}\cdot\frac{1}{x}$​
Multiplying by x gives:

$\displaystyle \frac{d\ \ln(f(x))}{d\ \ln(x)}=x\cdot\frac{d\ \ln(f(x))}{dx}$

Also by the chain rule:

$\displaystyle \frac{d\ \ln(f(x))}{dx}$
$\displaystyle = \frac{d\ \ln(f(x))}{d\ f(x)}\cdot\frac{d\ f(x)}{dx}$

$\displaystyle = \frac{f\,'(x)}{ f(x)}$

3. Aug 4, 2012

### cepheid

Staff Emeritus
It just means that you're taking the derivative of the function with respect to ln(x), rather than with respect to x. So, imagine you have a new variable y that is defined by the relation y = ln(x). So you're differentiating with respect to y here.

4. Aug 4, 2012

5. Aug 5, 2012

### Cinitiator

Does it also mean that d ln f(x) / d x is the slope of a log-lin plot?

6. Aug 5, 2012

### Dick

Sure. Sometimes called semi-log as well.

7. Aug 5, 2012

### Cinitiator

Thanks for your help. How is the practice of taking log-log plot derivatives called? Are there any general rules to follow? Is it vastly different from the normal differentiation?

8. Aug 5, 2012

### Dick

You are probably making too much of this. It probably not the sort of thing you see a lot of. Just relate them by the chain rule. SammyS already went thru that.