I found this lesson really confusing.
You say derrivatives of paraître, but how am I to know that appraître isnt a derrivative?
I understand your frustration but unfortunately, you will find exceptions to about every rule in French grammar so it is just a case of accepting this and just noting them and practicing them as often as you can.
Yes, it is unclear why it is not a derivative. Just one of those 'exceptions' to be learned. However, it is not at all clarified by the fact that these 3 sites conjugate apparaître (in passé composé) with avoir
Also, it nevers 'appears' on any list of the verbs that routinely conjugate with être.
Seems to be a couple of 'authoritative' references there that don't highlight it as being exceptional, indeed the Acadêmie notes its conjugation is as for connaître. (Having said that, some of the introductory language states it can also be conjugated with être, without a specific rule for this outlined).
For our purposes, I think the inclusion of apparaître in the title, especially without specific discussion is unnecessarily complicating and confusing.
And I can add, I just found this lesson which purports to list all the 'flexible' verbs that can use either être or avoir as auxiliary, depending on circumstance and meaning, at the end of the lesson. Apparaître is not included on the list, which together with the strong implication of this lesson that it does not use avoir, would suggest it was always conjugated with être. Monter can be used with avoir or être in Le Passé Composé depending on its meaning in French , nor is it found in this topic
(Apologies for the formatting but trying to copy the weblink just repeats the address for the "Monter" lesson).
More evidence that this lesson referenced in this question, largely because of its title and introduction, is probably incorrect and certainly confusing. I won't hold my breath for it to be fixed, given how long some of these issues have been being raised.
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