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:

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 , and is for . So, up to a constant multiple, it's like even for a "hugely" increasing function like

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 has **itself** s derivative: so that . More generally, has derivative 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, , so that which decreases as x increases- the amount of change stays constant while the value increases.