My personal opinion is that whilst a 'good' teacher should be making an effort to justify the topic of differentiation by explaining the limiting process involved, drawing chords and tangents etc, it's not necessarily useful nor practical to start testing students on doing things 'from first principles' at A level,(Original post by notnek)
A problem with teaching differentiation from 1st principles as an introduction to differentation is that C1 students have never studied limits (and most of them never will).
C1 should include a small section on limits - that would make 1st principles lessons easier for students as well as teachers.
One problem is that the topic of limits contains many subtleties, and once you get beyond simple polynomials there aren't many functions where you can actually apply first principles without getting into an awful tangle or going into a circular argument. (Every year I see people on TSR who've convinced themselves that they've found some new derivation for the derivative of e^x or ln x, when in fact when their argument is examined closely they've made some assumption about the behaviour of those functions and their associated limits. It's a bit similar to the people who think they're being clever using l'Hopital on the limit of (sin x)/x as x->0, not realizing that they need the derivative of sin x at 0 in order to be able to do this and this is precisely the limit they're trying to find!)
TL;DR: Yes, I agree the teacher should explain the process with a worked example for polynomials. But I'm not convinced of the merit of testing the student on this in a school exam
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- 08-09-2015 17:57