Does $f(dx)$ have any meaning? Simple question, does a differential $dx$ have any meaning composed in a function $f$, such as $\sqrt{dx}$, where $f(x)\neq x$?
Sort of. There is a dual concept to differentials, that of a "tangent vector", which is not unreasonable to think of as a kind of infinitesimal.

While $\mathrm{d}x$ is supposed to denote a differential, many unfortunately use the notation when they wish to speak of an infinitesimal. :(

Anyways, if $f$ is differentiable at a point $a$ and $\epsilon$ is an infinitesimal, then we have

$$ f(a+\epsilon) = f(a) + f'(a) \epsilon $$

Note this is a literal equality and not merely an approximation, as this kind variety of infinitesimal satisfies $\epsilon^2 = 0$.
Since $dx=\Delta x$, we may view $f(dx)$ as $f(\Delta x)$.