Since the tag derivations has no usage guidance at the moment, I do not know when it should be used, which is unfortunate.

6$\begingroup$ My old algebra memory says: a derivation on a ring $R$ is a map $D : R \to R$ satisfying $D(a+b) = D(a) + D(b)$ and $D(ab) = aD(b)+D(a)b$. Perhaps that is what the tag is for. $\endgroup$– Gerald EdgarJun 10 at 13:04

5$\begingroup$ @GeraldEdgar My concern is that some people might use it for questions on derivatives, i.e., as a synonym for differentiation $\endgroup$– Rodrigo de AzevedoJun 10 at 13:06

7$\begingroup$ I could also see it getting used for questions about logical derivations, i.e. formal proofs in the sense of natural deduction or similar systems. $\endgroup$– Peter LeFanu LumsdaineJun 10 at 14:42

$\begingroup$ Some stats related to the tag (derivations) can be found in the MO editors' lounge. Tangentially, this tag now has a tagexcerpt. $\endgroup$– Martin SleziakJun 28 at 9:45
2 Answers
I think that indeed:
in analytic questions about specific derivatives/ differentiation, the tag should be removed (it is then not very relevant: if it were, it would be relevant for thousands of questions). When necessary, it could be replaced by better tags (e.g. realanalysis, ca.classicalanalysisandodes, dg.differentialgeometry, differentialoperators... maybe also differentialcalculus although the latter doesn't sound like a researchlevel tag).
tag info should be added, indicating that this concerns derivations in a (kind of) algebraic setting.

$\begingroup$ Perhaps differentialfield would work for most algebraic uses? (I'm no expert but these are essentially the only algebraic derivations I personally have worked with. I understand that the concept is more general but we don't need a tag for every concept.) $\endgroup$ Jun 20 at 2:04

1$\begingroup$ Actually, I just learned from Wikipedia that differentialalgebra might be both more specific and comprehensive for the algebraic uses of the term. $\endgroup$ Jun 20 at 2:10

$\begingroup$ Re: differentialcalculus I agree that it's not really "researchlevel" but isn't that a little disparaging towards Newton, Leibniz, Gauss, Euler, Bernoulli, Bernoulli, Fourier, Bernoulli (is there a fourth Bernoulli?), Jacobi, Riemann, etc.? $\endgroup$ Jun 20 at 2:19

1$\begingroup$ @FrançoisG.Dorais Differential field is a field endowed with a derivation. This does not cover "derivation" in several respects: a derivation can be defined on a ring $R$ (not necessarily associative), and can also be defined from $R$ into an $R$module. Also, the set $\mathrm{Der}(R)$ of derivations on $R$ forms a Lie algebra and can be considered as a whole without singling out a single derivation. In important cases this is the Lie algebra of the group $\mathrm{Aut}(R)$, and occurs quite naturally. Quite many of the questions now with the tag would not be tagged "differentialalgebra". $\endgroup$– YCorJun 20 at 3:41

$\begingroup$ Yes, I am aware of the existence of such generalizations. I also believe that they are useful in contexts that I am not familiar with (or perhaps even some familiar contexts that can't remember at the moment). But this is mathematics as a whole, not actual MO usage. For tagging purposes, the tag name should ideally be both useful (in the MO usage sense) and unambiguous (when possible). The tag derivations is neither! $\endgroup$ Jun 20 at 4:12

1$\begingroup$ @FrançoisG.Dorais in any case, I agree that the tag differentialalgebra is better defined and should be used more often. It has no usage guidance. A draft: "Fields or rings endowed with one of several derivations. Differential Galois theory, algebraic study of ODEs/PDEs, etc." $\endgroup$– YCorJun 20 at 7:22
Another possibility is removing the tag altogether. Let us consider the pros and cons:
I can easily see people repeating the same mistake and misusing the tag, even with better usage guidance. So keeping the tag tidy will require further effort.
There seem to be only few questions that would keep the tag after the retagging YCor proposes, and it seems to me that they all contain the word "derivation" in the title. So it will be simple to search for questions on derivations even without the tag.

$\begingroup$ This may be a case where it is better to have a small amount of narrower tags than a tag for the overarching concept. $\endgroup$ Jun 20 at 2:07