## Natural log derivatives

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.
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Mentor
 Quote by 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.
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:

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

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

Mentor
 Quote by 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.
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.

Recognitions:
Homework Help

## Natural log derivatives

It's the slope of a log-log plot. http://en.wikipedia.org/wiki/Log-log_plot Plotting data like this is a way to discover a power law relation between f(x) and x.

 Quote by Dick It's the slope of a log-log plot. http://en.wikipedia.org/wiki/Log-log_plot Plotting data like this is a way to discover a power law relation between f(x) and x.

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

Recognitions:
Homework Help
 Quote by Cinitiator Thanks for your input. Does it also mean that d ln f(x) / d x is the slope of a log-lin plot?
Sure. Sometimes called semi-log as well.

 Quote by Dick Sure. Sometimes called semi-log as well.
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?

Recognitions:
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