Natural log derivatives

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.

SammyS

By the chain rule:

[itex]\displaystyle \frac{d\ \ln(f(x))}{dx}[/itex]

[itex]\displaystyle =
\frac{d\ \ln(f(x))}{d\ \ln(x)}\cdot\frac{d\ \ln(x)}{dx}[/itex]

[itex]\displaystyle =
\frac{d\ \ln(f(x))}{d\ \ln(x)}\cdot\frac{1}{x}[/itex]

Multiplying by x gives:

[itex]\displaystyle
\frac{d\ \ln(f(x))}{d\ \ln(x)}=x\cdot\frac{d\ \ln(f(x))}{dx}[/itex]

Also by the chain rule:

[itex]\displaystyle \frac{d\ \ln(f(x))}{dx}[/itex]

[itex]\displaystyle =
\frac{d\ \ln(f(x))}{d\ f(x)}\cdot\frac{d\ f(x)}{dx}[/itex]

[itex]\displaystyle =
\frac{f\,'(x)}{ f(x)}[/itex]

cepheid

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.

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.

Cinitiator

Thanks for your input.

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

Dick

Sure. Sometimes called semi-log as well.

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?

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.