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

Hi,

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

f'(x)/f(x)

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!

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

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

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

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

For complex variables, that expression takes on a new and different significance.

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 $\displaystyle f(x)= e^x$ has **itself** s derivative: $\displaystyle f'(x)= e^x$ so that $\displaystyle \frac{f'(x)}{f(x)}= 1$. More generally, $\displaystyle f(x)= a^x$ has derivative $\displaystyle f'(x)= (a^x)ln(a)$ 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, $\displaystyle f(x)= ax+ b$, $\displaystyle f'(x)= a$ so that $\displaystyle \frac{f'(x)}{f(x)}= \frac{a}{ax+ b}$ which decreases as x increases- the amount of change stays constant while the value increases.