Derivative to function ratio 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 $Q(x)<0$.

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.
