I was working on a Cosmology problem and got stuck at this approximation used in a paper. Fundamentally the approximation is, $\frac{df(x)}{dx} \ll f(x)$. Now I can't understand how to imagine this situation graphically, also what it means physically.
I tried to look up this question on the web and on this website but could not find anything close-by. Sorry if the question is badly phrased, I could not think of a better way to write it.
Any help would be appreciated.
It can be true or false, depending on $f(x)$. Note that $x$ must be unitless for the comparison to make sense. An extreme case would be $f(x)$ is a large constant-that would make the derivative zero. Large and almost constant would also satisfy the relation.
Note that rescaling $x$ can change this. If we define $y=100x$ we have $\frac {df(y)}{dy}=100\frac {df(100x)}{dx}$ and the $\ll$ might not be satisfied.