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

• Oct 29th 2012, 05:07 PM
visserthree
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!
• Oct 29th 2012, 05:24 PM
johnsomeone
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.
• Oct 30th 2012, 09:00 AM
HallsofIvy
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.