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Natural log derivatives

  1. Aug 4, 2012 #1
    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. jcsd
  3. Aug 4, 2012 #2

    SammyS

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    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]
     
  4. Aug 4, 2012 #3

    cepheid

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    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.
     
  5. Aug 4, 2012 #4

    Dick

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  6. Aug 5, 2012 #5
    Thanks for your input.

    Does it also mean that d ln f(x) / d x is the slope of a log-lin plot?
     
  7. Aug 5, 2012 #6

    Dick

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    Sure. Sometimes called semi-log as well.
     
  8. Aug 5, 2012 #7
    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?
     
  9. Aug 5, 2012 #8

    Dick

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    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.
     
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