Derivative to function ratio

Question

Is there a physical meaning for derivative of the function to function ratio? That is, this quantity, $$ Q(x) = \frac{1}{f(x)}\frac{df(x)}{dx} $$ Like for instance, if $f$ is the potential energy, this would be work to potential energy ratio. Or even what can we say about $Q(x)$, say when $Q(x)<0$.

Answer 1

This would give $$ \int\limits_{x_0}^x \!Q(\xi) \, d\xi = \ln \frac{f(x)}{f(x_0)} \iff \\ f(x) = f(x_0) \exp {\int\limits_{x_0}^x \! Q(\xi) \, d\xi} $$ An example would be a radioactive decay process ($Q$: decay constant)

Another: light absorption ($Q$: attenuation coefficient).

Answer 2

It is also $\frac {d(\ln(f(x))}{dx}$. Note that it has units of the units of $f$ divided by length, so it can be a scale length that shows how quickly $f$ changes.

Answer 3

I know in Physics they use that $\frac{dN}{dt}=Nk$ for various uses so you could use it to show whether a relationship is exponential or not

Answer 4

If $f(t)$ measures the size of the population, and $t$ is time, then $Q(t)=f'(t)/f(t)$ is the rate of change of the population per capita.

For example, if each individual reproduces at a constant rate $r$, then $f'(t)/f(t)=r$, so we get exponential population growth $f(t)=f(0) \, e^{rt}$.

But if the per capita growth rate decreases linearly with the size of the population (due to limited resources, for example), $f'(t)/f(t)=r(1-f(t)/K)$, we get logistic growth.