How well are you taught calculus?

I can do a few extra things like first principles, induction proofs and some work on the fundamental theorem of calculus, but these were largely self-taught, school couldn't give a damn about more maths

Differentiation from first principles is covered in C1, isn't it? Likewise, proof by induction is in FP1.

I'm shocked that your school didn't even teach you the few parts that are actually on the syllabus.

Are you sure? Although the C1 book does skip over proving the power rule, it seems to have a solid explanation for differentiation from first principles. There are around 5 pages devoted entirely to it.

isn't that more to do with the differentiation you come across in C1 though, rather than more complex stuff such as trig and whatnot

by not on the syllabus I mean not actually tested in the exam

As someone who has been self-learning A Level Maths and Further Maths entirely from the internet and from the official Edexcel textbooks, I've constantly been bashing my head against the wall with the shocking lack of emphasis placed by the syllabus on actually understanding calculus.

Although there are a few proofs for the various methods of differentiation, the syllabus, by and large, completely skips over many insights and proofs -- very few of which would require degree level maths to learn.
One such example is the fundamental theorem of calculus which links integrals as the sum of infinite rectangular strips representing the area under a curve with the process of antidifferentiation. Without it, you're left thinking that via some process of magic, finding an antiderivative of a function will tell you the area under a curve.

So, what I want to ask is, for those who are studying at school, are you taught any concepts and proofs for calculus that are outside the syllabus?

Edit: By not on the syllabus, I mean not covered in either the syllabus specification or in the official textbooks.

Differentiation from first principles is covered in C1, isn't it?

Proving the fundamental theorem of Calculus really isn't that easy and requires a a decent bit of analysis.

I learnt differentiation from first principles at school and it came up in my exam

Not in all specifications. OCR, for example, says knowledge of first principles is not required.

It might say that, but does it show you anyway? If so, I'd consider it a part of the syllabus.

How do you mean "does it show you anyway"? The specification is just a list of topics that may be examined and this one is precluded. If it isn't examined few teachers will teach it. Assessment drives the curriculum.

Although there are a few proofs for the various methods of differentiation, the syllabus, by and large, completely skips over many insights and proofs -- very few of which would require degree level maths to learn.
One such example is the fundamental theorem of calculus which links integrals as the sum of infinite rectangular strips representing the area under a curve with the process of antidifferentiation. Without it, you're left thinking that via some process of magic, finding an antiderivative of a function will tell you the area under a curve.

In the official textbooks provided by OCR.

Official textbook is an overstatement. There is an "approved" textbook from Cambridge University Press (who are in common ownership with OCR) and that includes it but many schools do not use this semi-official textbook. One good thing Michael Gove is doing is banning links between Awarding Bodies and publishers.

Well, if you want it "properly" then no, you couldn't really skip 'degree level' stuff; i.e., content one gets in an Analysis course.

For the Fundamental theorem, it's proof (I'm going by the one on wiki as a reference - the first part) uses the mean value theorem for integrals, which is not a level spec. There's an easy way to see it's true (geometric reasoning), but that wouldn't be proper; a proof would (usually, there of course could be other proofs) use both the extreme value theorem and the intermediate value theorem. The EVT and IVT are 'obvious enough' to be taken for granted, but again, that wouldn't be proper. You see that this is beginning to go down a rabbit hole!

It's tricky to say in how much depth one should go for in an A level course. At the end of the day, most students will use calculus as a tool at most; generally, only those who pursue a maths degree will care about the details, and they certainly will get them. It's good to have some depth of understanding, but of course it should be noted that calculus was essentially reinvented in the 19th century to completely formalise its arguments. An A level student seeing calculus for the first time is likely to need some time to get used to the concepts how Newton/Leibniz thought of them, let alone Bolzano/Weierstrass/Cauchy/Riemann etc.!

I agree that most students will have little use for doing a full on uni course on calculus, but the goal should be to have them understand the calculus they currently learn. All the minor details like the proof for the mean value theorem can be left for uni. It's intuitive enough that demonstrating the concept is enough to gain understanding and be able to use it to prove the fundamental theorem of calculus, which is not intuitive.

I'm not sure students really need much time to become comfortable with the concept of limits, differentiation from first principles, Riemann sums, Newton/Leibniz notation, etc. Going by my own experience, which was greatly hampered by having no teachers to ask questions of, 6 months should be enough time to learn all of calculus in the maths A Level, including proofs for things like the fundamental theorem of calculus.