Choosing Notation for Variable Substitution into Derivative Expressed with Differentials

Question

Consider function $f(x)$. I've counted 4 possible notations to write a derivative of $f(x)$ at point $x = a$:

$f'(a)$;
$\frac{\operatorname{d}{f(a)}}{\operatorname{d}x}$;
$\left.\frac{\operatorname{d}{f(x)}}{\operatorname{d}x}\right|_{x = a}$;
$f_x(a)$ in case of partial derivative.

Currently, I need to use the one which involves differentials, i.e. either #2 or #3. I've listed all of them for the sake of completeness because I want to draw a point and find out your opinion on it.

The point is that for me, personally, #1, #3, and #4 look perfectly fine. However, #2 looks ambiguous. In other words, I could interpret it in 2 ways:

Take derivative of $f$, and then substitute $x = a$;
Substitute $x = a$, and then take derivative of $f$ (what would be stupid in most cases, but still).

That's why I'm inclined to prefer #3 throughout my paper. So, the questions are:

What do you think about all of this and my argumentation?
How common is #2 in mathematical literature?
Which one of #2 or #3 would you recommend?

More common than either of #2 and #3 is $\frac{\mathrm{d}f}{\mathrm{d}x}\vert_{x=a}$. — Dan Petersen, May 12 '14 at 13:11
$\frac{\mathrm{d}f}{\mathrm{d}x}|_{x=a}$ might indeed be more common (it's even part of the ISO 80000-2 standard), but it doesn't make any sense under the modern interpretation of $f$ as a map (say of type $\mathbb{R}\to\mathbb{R}$), since "$f$ doesn't know about the name of its input variable". It's very plausible that historically Jacobi contributed to this misunderstanding, although the $f$ in his article was of type $\mathbb{R}$. — Michael Bächtold, Sep 06 '17 at 11:32

score 2 · Accepted Answer · edited May 12 '14 at 13:54

#2 seems ambiguous, since it seems, at first glance, to be the derivative of the constant $f(a)$, i.e., the interpretation of #2 is usually the first interpretation in the second list in the question. #3 is the most common, and is used in almost all calculus textbooks and/or papers to denote the derivative of $f(x)$ evaluated at $a$. I would recommend #3, since that is how I have seen "the derivative of $f(x)$ evaluated at $a$" written.

Choosing Notation for Variable Substitution into Derivative Expressed with Differentials

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