I think the reason for this is that derivation from 'first principles' isn't really done. You'll do it once or twice in the intro to derivatives and that's it.

I think the reason for this is that derivation from 'first principles' isn't really done. You'll do it once or twice in the intro to derivatives and that's it. The other 40 hours you spend on derivatives won't even touch it. The issue with being able to derive the formulas for derivation yourself is that it's not very useful. You simply don't have time to make those derivations during a test. It's like trying to use grammar rules in a conversation - conversations happen at a pace where you cannot apply grammar rules. You'll just have to know the patterns. You learn things in school to do a test. The usefulness of the vast majority of the knowledge they attain is purely to help them do the test. Later in life you might wish you knew more about this or that, but that's not at all apparent to the student.