I think you are overreacting a bit. A "morass of missing definitions" is not the same thing as being meaningless. One can provide the missing definition in term of the Frechet derivative, but it is not clear from the question that this would be at a level appropriate for the OP. What the answer you criticise does is explain why OPs argument does not work and therefore answer the actual questions: "So something is very wrong here, no? Can anybody explain to me what just happened?"

@MichaelGreinecker I agree with you with respect to the appropriateness of level, abstractly speaking. But I can't honestly look at the comment and think this particular case is just an issue of restricting the "level" of an answer. Even if it were, what is usually done is "Ah, you can use the chain rule, but this way is more elementary". or "I personally don't know how you would apply the chain rule, but this way is more elementary". Or simply "This way is more elementary". That prelude and comment have entirely different connotations (compare to the manufactured answers I gave).

Furthermore, let's suppose that this is a matter of level only (which, as I said above, I disagree in this particular instance). OP explicitly mentions the chain rule and an attempt at its use. The question is "So something is very wrong here, no? Can anybody explain to me what just happened?". Would it be really justified to disregard something which someone is trying as "beyond his level" when he explicitly mentions that he is willing to understand it (and why it didn't work)? (not only that, again: the question is precisely why it didn't work. He even repeats that in a comment.)

@GerryMyerson I understand. However, the focus is on the "ignored by many" part. That said, what is "disagreement" about the chain rule? How can one "disagree" with a theorem?

To all: please, you can criticize my comparisons, say that I'm overreacting etc. That is all fair, really. What makes me sad is that that is being the focus, and not the objective problem at hand. Do you seriously think that what is more important to the site, as a whole, are comparisons made in a meta post in a subjective fashion instead of, I repeat the term, a negligent spread of misinformation?

Say someone is learning analysis, and wants to compute the derivative of a map (say the map in said post) using the chain rule. He is getting trouble, and is not managing to do so succesfully. Something does not feel right. He comes to MSE expecting some enlightment. Then he stumbles in that question, and has an answer by a respected user that has been "approved" by the community which says "hey, what would what you are trying to do even mean? Forget it, here is what we can do." I can't understand, for the life of me, how can, after trying successively (and being ignored) to rectify this (...)

(...) the community feel that I am overreacting (which is what I feel it is communicating to me by the number of downvotes). I've been respectful in all my interactions and, at the end of the day, the only thing I want is that the situation above does not happen. Like I said, erasing that first line (and the comment) would solve the issue. It saddens me deeply that after all this, one of the comments (with 6 upvotes as of now) is one that is essentially dismissing every point I made and the above aspect of this situation to "hey, Godwin's law."

@AloizioMacedo The chain rule for functions in one variable is plainly not the same thing as the chain rule for Frechet derivatives and one can prove the former in a way that does not generalize to the latter. And the chain rule for functions in one variable, which OP clearly tried to apply, does not apply to the problem under consideration.(...)

And while explaining that a chain rule for a more general notion of a derivative is the appropriate one and does apply is a legitimate way to answer the question, so is explaining that the chain rule for functions in one variable does not apply and isn't even meaningful in this context. I hope this answer what you call the "objective problem".

@MIchaelGreinecker First of all, thank you very much for bearing with me here. I sincerely appreciate it. That said, I can't help but feel that your justification is ad hoc. I don't think that is what the answerer had in mind (which is unfalsifiable, but still relevant to be explicit). And you understand the whole mathematical context. It is easy for you to phrase and conceive what is being said in an intelligible way. Now try to picture the situation as a student which is struggling to understand it. They are who matter here. Do you seriously think that answer/comment is appropriate?

@AloizioMacedo It answered the question and OP seemed to be happy with it. I think you could have started your answer by saying that even though the version of the chain rule OP is familiar with does not apply, there is a way to make this rigorous. I think "the morass of missing definition" is for me an indicator that it is too much of an effort to fill in the blanks to make this work in the context of the question.(...)

Whether this is true will depend on a number of things, mostly on whether the idea of the derivative as a linear approximation is familiar to the OP. But I see no point in the drama, just two people disagreeing on what is pedagogically appropriate here.

@MichaelGreinecker I don't understand why you think that OP is not familiar with the chain rule. Mistaking the derivative of $X^2$ for $2X (\cdot)$ instead of $X(\cdot)+(\cdot)X$ is an honest "newbie" mistake, and which does not entail at all complete non-understanding of the chain rule. Why should I (and why did you) assume that OP does not know what he is asking about? If I have trouble computing the integral of $f(x)=x$ on $[0,1]$, and want to know what am I doing wrong, is it reasonable to conjecture that I know nothing about integration, and should note that it is the area of a triangle?

The OP explicitly in their notation only took derivatives with respect to t. And, presumably, they did not learn about Frechet derivatives in a course, learned the chain rule in that context, and accepted the answer given. Also, HM did ask in a comment what the intended definition of the derivative is and OP did not answer, which indicates they had no answer for the question.

"What would such a derivative even mean?" doesn't look like a genuine inquiry. And he only took derivative with respect to t because he is wishing to compute a composition which has as domain $\mathbb{R}$, I don't see the issue there. Your justification "And, presumably, they did not learn about Frechet derivatives in a course, learned the chain rule in that context, and accepted the answer given" is sound, but a posteriori (the issue here is the answer itself).

As a counterpoint to the a posteriori argument with also an a posteriori argument, this question: math.stackexchange.com/questions/2228111/pullback-connection suggests to me that there is great chance that he indeed knows the frechet derivative at least in the finite dimensional case.

(the argument being, I don't know someone who studies differential geometry and has not seen the frechet derivative). That said, I retiterate that I think this is beyond the point.

What OP really knows if kinda irrelevant when judging whether the behavior of a person answering the question was appropriate. Indeed, that is why many of us care about providing context in question- as OP did not do. So there are two ways to clear up the confusion of the OP: Either explain why the naive application of the introductory calculus chain rule does not work, as HM did, or explaining why the appropriate version of the chain rule actually does work, as you did.

If you think the first version was inappropriate, making a comment was enough and I don't see the point of the drama on meta. HM's answer did a nice job of explaining why OPs approach was doomed to fail and made the point in a very nice elementary way. Presumably, that is why the answer was upvoted and accepted. And, yes I think HM could have chosen his words better. I also think you could have chosen your words better.

I see. Thanks again for your attention. I agree that the way the post was phrased on meta had a dramatic undertone, but these are things which hit me personally. Although I see your points, I will be honest: I can't shake the feeling that you are taking the best possible interpretation to the answer. For instance, you were arguing at first based on the level of OP, and then you shifted opinions saying it is kinda irrelevant now.

But I think we've reached enough common grounds, and I truly appreciate the conversation. One last thing (I need to go in a few moments), where exactly I could have chosen my words better? In the answer to that post proper, or in meta, or here? If in all of those, still, what words exactly?

Something that is indeed true is that I interpret the answer in a rather benevolent way- which might not necessarily be correct. It is here why I think you could have chosen your words better. You could have written your answer and your comments as providing a different perspective instead of as a criticism of an other answer. Reading your answer again, it was actually fairly neutrally worded, unlike the meta post. The meta post was way too adversarial IMHO.

I agree that the meta post is bad in one way or the other. I tried to refrain myself from being "adversarial": I explicitly did not mention the user's name, and I tried not to refer as little as possible. Believe it or not, I have no grudge with him whatsoever. I believe the "dramatic" assessment is more appropriate that adversarial. That said, I ask for a little empathy with respect to the drama: I was ignored, for over half an year, in different ways, with respect to an issue which

hits me personally. With the fear of being repetitive, you truly don't understand how it is comforting to be listened. I don't think I was being unreasonable in any thing I did up to now, but I never got a response. From anyone. Only mild upvotes which may means something, but... well, they are not words. And even after I post it, one response (the only other, aside from yours) is an allusion to Godwin's law.

I can empathize with the frustration, I often see answers that are not really wrong but IMO pushing a misleading perspective. But you should also empathize with people here having similar experiences on this humongous site and having to deal with similar problems. If everyone facing some minor issue makes a big production on meta, meta would be flooded completely. So people tend to react negatively to what seems to be making too much of production out of something small.

The allusion to Goodwin's law was appropriate IMHO. The power of the poem certainly relates to the grave and horrible things people did not speak out against and even if someone would be wrong on the internet somewhere, itis just not on the same level.

Anyways, I have to leave and take care of some stuff. I think talking to you was productive.

If I may add, it was a bit unclear to me whether the question is mainly about these two specific answers or whether it is asked as a general question. (But the reason might be that the comments concentrated mainly on this specific case.)

I am a bit surprised by the number of downvotes, since it is a legitimate issue to ask. (Not everybody will agree with your viewpoint, but that does not change that fact that asking about quality of answers is a reasonable question.)

I would not be surprised to see some delete votes on that question. (Recently quite a lot of questions on meta have been deleted, some of the deleted ones I considered to be reasonable questions. If meta starts working in the way that people disagreeing with something simply delete the post instead of replying, it is probably not good.)

I vaguely recall that quid also mention deletions on meta in some discussion but now I cannot find it.

There is one specific thing in your post which I am not sure is accurate: "Why do we put so much scrutiny over newcomers, and so little over experienced users?"

Maybe I am biased because I see many deletions on meta. But certainly also answers of high-rep users are being deleted.

Just notice this message as an example. (I would probably not directly name names as an example of this in comments on meta, but probably here in chat this is less visibile.)

@Martin Sleziak The question is about an issue which I've only seen manifest in these two specific answers. Were they existent in the same egregious way (that is, an intersection of: ignoring requests to rectify, upvoted, by a user of outstanding reputation, ignored by those who could help get the point across etc) in other instances, I would certainly link them. The problem is general in abstract. Unfortunately (or rather, fortunately), I am only aware of those two under those conditions.

So yes, with respect to your first point, it is asked as a general question.

With respect to the specific thing you mention, I don't know about Michael, but the other user is mentioned recursively in meta if I recall correctly. He has a "history" of bad answers (just notice how Did refers to them). This, I think, is an indirect argument to my point: it seems that history is heavily considered when upvoting and not downvoting, and also when simply considering. The comment by Did itself is evidence of that.

Simply put, let me rephrase and update my point: where I say "experienced", consider "well-established in the community". Instead of "newcomers", read "less-established", which has as a subset newcomers. I think this phenomenon is, unfortunately, natural. The extent of it is what we should worry about, and the neglect of the community to even get to that point of the discussion which I tried to raise tells me that we may indeed be in a critical point.