What is the significance of taking derivatives in the form d ln f(x) / d ln x?

[Cinitiator]

Homework Statement

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?

Homework Equations

d ln f(x) / d ln x

The Attempt at a Solution

Googling.

 

[SammyS]

Homework Statement

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?

Homework Equations

d ln f(x) / d ln x

The Attempt at a Solution

Googling.

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]

Homework Statement

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?

Homework Equations

d ln f(x) / d ln x

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.

 

[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]

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.

Thanks for your input.

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

 

[Dick]

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.

 

[Cinitiator]

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?

 

[Dick]

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?

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.