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Given a function $f(x)$ and its derivative $f'(x)$, is there a term for $\frac{f(x)}{f'(x)}$ or for $\frac{f'(x)}{f(x)}$?

Bitwise
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    Not sure if this is the kind of thing you're after, but the latter is referred to in finance as the "interest rate" :-) – Steve Jessop Jun 04 '15 at 21:38
  • @SteveJessop based on the answer, does the interest rate have something relate to the logarithm function? – Ooker Jun 04 '15 at 23:11
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    @Ooker: not exactly by definition, but yes it does. Consider an exponential function $f(x) = e^{Ax}$, then the "interest rate" is $A$, it controls how steeply the curve climbs. And $log(f(x)) = Ax$, lo and behold, has derivative $A$. It also turns out (by the chain rule), that the result holds in general (quoted by muaddib), not just for the exponential function. – Steve Jessop Jun 04 '15 at 23:23
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    The intuition here is that the logarithm "turns multiplication into addition". Therefore the rate at which the logarithm is increasing, is the rate at which the function is multiplying, i.e. the rate of compound interest. Add to the logarithm of the function = multiply the function. Sort of thing. – Steve Jessop Jun 04 '15 at 23:25
  • In economics, the latter is closely related to the Arrow-Pratt measure of absolute risk aversion of a utility function: $A(x;f)=-\frac{f''(x)}{f'(x)}$; if we take @muaddib's notation and modify it slightly, we get the (equivalent analog to the) Arrow-Pratt-De Finetti measure of relative risk aversion: $\frac{d\log(f(x))}{d\log(x)}=\frac{xf'(x)}{f(x)}$, while $R(x;f)=-\frac{xf''(x)}{f'(x)}$ – MichaelChirico Jun 05 '15 at 17:41

1 Answers1

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Certainly the latter one is called the logarithmic derivative: $$\frac{d\log(f(x))}{dx} = \frac{f'(x)}{f(x)}$$

muaddib
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