What is the significance of using "$a$" vs "$x$" in this text?

I'm a web development guy currently learning Calculus and am having some trouble understanding the seemingly unwritten rules of variable naming conventions in mathematics.

I've read several other questions relating to "variable naming conventions" here (e.g. laziness on the part of an equation's author, variables are transient, meaningless values, etc.), but I'm still not understanding the motivation driving certain variable name choices.

For instance, I've found certain passages in my various textbooks in which there is an apparent significance in the choice of name an author assigns to a variable. Being unable to understand the thought process involved in examples like the one below is a bit of an impediment to my comprehension of the concepts that are being conveyed.

As a specific example, here is an excerpt from my current textbook, "Calculus: Early Transcendentals," 7th edition, by James Stewart (the guy who built the Calculus house):

In the preceding section, we considered the derivative of a functon f at a fixed number a. $$ f'(a) = \lim_{h \to 0} {f(a+h)-f(a)\over h} $$ Here we change our point of view and let the number a vary. If we replace a in Equation 1 by a variable x, we obtain $$ f'(x) = \lim_{h \to 0} {f(x+h)-f(x)\over h} $$

Here, Stewart evidently has some internal distinction between a and x, but it was not mentioned until this point, at which he gives virtually no explanation other than to say that a is a fixed number.

But it looks like a variable, not a fixed number. I get that he is talking about the derivative of a function at a fixed number, like 3, as opposed to a derivative of a function over a range of numbers, but for someone with a software development background, like me, the seemingly arbitrary switch in variable names is confusing.

Is there a somewhat universal convention in mathematics for choosing variable names in different contexts? Or is it just a matter of individual preference, with an explanation of the choice left to the individual author of the equation/formula/expression?

Any help to shed light on this confusing topic would be very much appreciated.

In mathematics, $x$ usually denotes a variable and $a$ denotes a (fixed) constant (however, any constant).

The idea that the author wanted to give is that if you can calculate the derivative at any point, then you can consider the function that sends each point $x$ to the derivative of $f$ at $x$ (function known precisely as the derivative of $f$).

There's no real difference between $x$ and $a$ used as variable names. We distinguish between variables, parameters, constants - and yet this distinction does not really exist. This hierachy is just a customary order among variables, as it is customary to denote "more variable" variables with $x$ and "more constant" variables with $a$, say. Or to use $n,m,k$ and a few others for integers. Indeed, we could have called the function $x$ and its parameter $f$ without producing anything wrong - it just becomes somewhat illegible due to the readers expectations: $$ x'(f)=\lim_{n\to 0}\frac{x(f+n)-x(f)}{n}$$

To give an example that is closer to "your world" maybe: The hostname of a web server need not start with www and that of a mail server need not start with mail. Yet it is customary to run weservices on www.example.com and mail transfer on mail.example.com instead of the other way round. It certainly works fine the other way round but people might get confused.

Note that, perhaps somewhat counterintuitively, there is no actual difference between "a fixed number $z$" and "a variable $z$". Both mean exactly the same thing: for an arbitrary element $z$ in the domain of $f$, $f'(z)$ is defined to be such-and-such real number. Logically "fixed numbers" and "variables" have the same semantics and correspond to universal quantification (for every $z$ we do the following / the following holds).

The difference is in pragmatics, that is, in what purpose we want to signal to the reader (you can perhaps think of this as defining multiple type synonyms for the same underlying type - you make up creative names for basic types to, among other things, better describe what the code does).

"A fixed number $a$" signals that the reader is supposed to concentrate on values $f'(a)$ one at the time, and the choice of name reinforces this wording. "A variable $x$" means that the reader should think about $f'(x)$ changing as $x$ does, and about the function $f'$.

As for the appropriate choice of names, it's impossible to give hard-and-fast rules, but you get used to various conventions subconsciously, and others have already given some useful guidelines.

edit: I overlooked that Hagen von Eitzen's answer makes the same point about variables and constants being technically the same thing, so this is mostly a duplicate of his answer.

If you think back to the time when you were in 9th grade learning to solve quadratic equations, what you saw was that $$ \text{if } ax^2+bx+c=0\text{ then }x=\frac{-b\pm\sqrt{b^2-4ac\,{}}}{2a} $$ (not to be confused with $\dfrac{-b\pm\sqrt{b^2-4a} c}{2a}$ or $\dfrac{-b\pm\sqrt{b^2-4} ac}{2a}$ or $\dfrac{-b\pm\sqrt{b^2-{}} 4ac}{2a}$, etc., all of which I've seen frequently $\ldots$)

And in individual problems you'd see things like $$ 2x^2-11x + 9 = 0,\quad\text{so that }a=2,\,b=-11,\,c=9. $$ Thus $a,b,c$ were the things that were known in each particular instance and $x$ was not.

However the way Stewart does this is silly, for reasons pointed out in the original question above.

Here's my favorite example of the fact that understanding the concepts of "constant" and "variable" can matter, and that it depends on context: \begin{align} \frac d {dx} 2^x & = \lim_{h\to0} \frac{2^{x+h}-2^x} h & & \text{(This step you should know by reflex.)} \\[10pt] & = \lim_{h\to0}\left( 2^x \cdot \frac{2^h - 1} h \right) & & \text{(This step is routine algebra.)} \\[10pt] & = 2^x \lim_{h\to 0} \frac{2^h-1} h & & \text{(because $2x$ is }{\bf constant}) \tag 1 \\[10pt] & = \left(2^x \cdot\text{constant}\right) & & (\text{because the limit is }{\bf constant}). \tag 2 \end{align}

Note:

• In line $(1)$, saying $2^x$ is "constant" means it doesn't change as $h$ changes.
• But in line $(2)$, saying the limit is "constant" means it doesn't change as $x$ changes.

"Constant" means not changing as something changes, but what the "something" is can depend on context!

BTW $\dfrac d{dx} e^x = \left(e^x\cdot\text{constant}\right)$, just as above, but the thing that is "natural" about $e$ rather than $2$ as the base is that when the base is $e$ rather than $2$, then the constant is $1$.