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Math Help - Interpretation of f'(x)/f(x)

  1. #1
    Apr 2012

    Interpretation of f'(x)/f(x)


    I was just wondering how one could interpret in words in describing the ratio of a derivative over its original function as follows:


    I'm having a bit of trouble trying to interpret what this function is saying especially as x is changing.

    Any help is much appreciated!
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  2. #2
    Super Member
    Sep 2012
    Washington DC USA

    Re: Interpretation of f'(x)/f(x)

    That expression is sometimes refered to the logarithmic derivative, because:

    \frac{f'(x)}{f(x)} = \frac{d}{dx} \ln(|f(x)|)

    Since the logarithm grows very slowly, that expression - the rate of change of ln(|f(x)|) - should be "small" expect where a function is really "spiking" - growing or shrinking very quickly.

    Consider, it's 1 for e^x, and is \frac{n}{x} for x^n. So, up to a constant multiple, it's like \frac{1}{x} even for a "hugely" increasing function like f(x) = x^{1000}.

    For complex variables, that expression takes on a new and different significance.
    Last edited by johnsomeone; October 29th 2012 at 06:52 PM.
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  3. #3
    MHF Contributor

    Apr 2005

    Re: Interpretation of f'(x)/f(x)

    You can also think of that as the 'relative' change, amount of change relative to the value of the function. The reason that this gives a "logarithmic" derivative is that f(x)= e^x has itself s derivative: f'(x)= e^x so that \frac{f'(x)}{f(x)}= 1. More generally, f(x)= a^x has derivative f'(x)= (a^x)ln(a) and [tex]\frac{f'(x)}{f(x)}= \frac{a^x ln(a)}{a^x}= ln(a), a constant. On the other hand, for a linear function, f(x)= ax+ b, f'(x)= a so that \frac{f'(x)}{f(x)}= \frac{a}{ax+ b} which decreases as x increases- the amount of change stays constant while the value increases.
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