TITLE: In the derivative notation $f'(x)$, does the $(x)$ mean "with respect to $x$" or something else?
QUESTION [5 upvotes]: I know $d/dx$ means derivative with respect to x (perhaps I am a little unclear on what precisely that means - I'm not quite sure - but I do think I have at least some sense of its meaning). I know (I think!) that... $$\frac{d}{dx} f(x)=f'(x)$$
Here is my question (it's been bugging me incessantly for the last couple weeks): What does $f'(x)$ mean? Does it mean the derivative of $f$ with respect to x? Or, rather, does it mean the derivative of $f$ with respect to x, evaluated at the point x? Or something totally different? 
To make my question more clear, let me ask this too: What does $f'(a)$ mean? Does it mean the derivative with respect to $a$? Or the derivative with respect to $x$ evaluated at some point $a$? (so that the "with respect to x" is actually "encoded" in the $f'$ part of the notation!). Etc. Even worse, take $f'(ax)$, which appears in some derivative computation rules. Does that mean derivative with respect to $ax$?? 
I suspect this confusion may be somewhat related to the maddeningly persistent confusion -- both for me and virtually everyone else -- between a function and its value at a point. Also, teachers tend to use somewhat imprecise notation and language, so I, being someone who likes precision, can sometimes get confused. 
While I'm at it, I'll note that this confusion may be related to my additional confusion over language like "the derivative of the sum of two functions," where the two "functions" are, say, $x^2$ and $x^3$. But I thought those were mere polynomials, not functions. One might say $f(x)=x^2$, but one would never say the function itself, $f$, was equal to $x^2$, right? I'm confused. We shouldn't say things like "the function $x^2$," right? :(

REPLY [2 votes]: When declaring a function we need to specify 3 things: a domain, a codomain and a mapping rule. In particular, when $f$ takes only one variable as input, we declare it as
$$f\colon A\to B, \quad f(x)=\text{some expression in }x.$$
The symbol $f(x)$ represents an element of $B$, but not the function itself. The symbol $f$ is the function. The symbol $x$ is irrelevant here, we could have used $\diamond$ if we wanted to.
We can obviously make functions that take functions as input and produce functions as output. Such functions are sometimes called operators; the derivative is an example of an operator, called the differential operator, and both $d/dx$ and $'$ is what we use to denote it. The fact that $x$ appears here is again irrelevant, we could have used $\frac{d}{d\diamond}$ if it was clear from the context that we use $\diamond$ to represent the indeterminate.
As $d/dx$ is a funciton, it has a domain (of which e.g. real valued differentiable functions in one variable are elements of) and a codomain. So if $f$ is in its domain, then $\frac{d}{dx}f$ is the corresponding element in the codomain, i.e. the output is a function in the codomain, and not a function in the codomain evaluated at some point. Since $'$ represents the same operator, $f'$ represents the same element in the codomain as $\frac{d}{dx}f$.
So what is $\frac{d}{dx}f(x)$? It is $\left[\frac{d}{dx}f\right](x)$, i.e. the function $\frac{d}{dx}f$ evaluated at $x$. Similarly, $f'(x)$ is the function $f'$ evaluated at $x$.
Then, what does $f'(a)$ mean? The same thing as $f'(x)$, except we replace $x$ by $a$. But be careful: $f(a)'$ doesn't mean the same thing.
About you other concern: first, polynomials are functions. Communicating mathematics can be very different to doing mathematics. When we say that a function is $x^2$, it is implicitly understood that we are talking about the square function, i.e. the real-valued (also, possibly complex-valued) function $f$ in one real (or complex) variable such that $x\mapsto x^2$. But after a while it gets really tedious to keep repeating all of that, when we all know that "square function" or "the function $x^2$ is to be understood as the same thing.