How can I best improve my understanding on linear operator interactions: $df/dx$ vs $df(x)/dx$, and $df/dx$ vs $dy/dx$, etc? For years now, I had always assumed that $\frac{df}{dx}$ was notational shorthand for $\frac{df(x)}{dx}$, because many teachers and authors of papers treat them interchangably. However upon getting studying deeper into vector/matrix calculus, and the general general theory of operators, I've realized that $\frac{df}{dx} \ne \frac{df(x)}{dx}$, likewise $\frac{df}{dx} \ne \frac{dy}{dx}$, and sometimes $\frac{df(x)}{dx} \ne \frac{dy}{dx}$. IE, the difference between $df$ and $df(x)$ is that the latter essentially involves a composition operator, which technically changes the logical/mathematical meaning of the expression. This is likewise further complicated by the fact that some authors use $d$ and $\partial$ interchangably, whereas others use them to indicate different properties of the differential.
While I able to observe a general pattern to the identities and operator interactions in terms of associative/commutative order of operations, like with integration by substitution and differentiation rules, they don't seem to be completely consistent, which means there's something I am not understanding.
TL;DR: I am basically trying to get a clear intuition on the interaction/relationship between the composition operator and differential operator, but cannot really find any sources that elucidate this - and such is further complicated by the fact that many authors seem to abuse notation using forms interchangably.
Note: I definitely don't know everything about operator theory, as there is still a lot I am learning, but I can't find many good learning resources about the topic.

Generally we don't distinguish much between  $\frac{\mathrm{d}f(x)}{\mathrm{d}x}$ and $\frac{\mathrm{d}f}{\mathrm{d}x}$, but formally speaking we can view $\frac{\mathrm{d}f(x)}{\mathrm{d}x}$ as the value obtained when evaluating the derivative of $f$ at a point $x$, i.e
$$\frac{\mathrm{d}f(x)}{\mathrm{d}x}=f'(x) = \lim_{h\to 0} \frac{f(x+h)-f(x)}{h}$$
Whereas we can view $\frac{\mathrm{d}f}{\mathrm{d}x}$ as the function defined as the derivative of $f$, i.e,
$$\frac{\mathrm{d}f}{\mathrm{d}x}=f' = \mathcal{D}(f): x \mapsto  \lim_{h\to 0} \frac{f(x+h)-f(x)}{h}$$
Here $\mathcal{D}$ denotes the differential operator.
