Background: I am new to this site, but have 1500 reputation on the main Maths Stack. I am (age-wise) a secondary student of maths, but for a very long time have been informally learning at home and been studying linear algebra, calculus and formalism that goes far beyond the A-level and Further Maths curriculums. “A-level”, for non-UK readers, means people from 17-18 years old, just starting to learn calculus but without any real formal background - I’ve seen the textbooks and they really just state facts and don’t explain them! I have been helping my friends with their maths for a long time, and have acted as an informal tutor to them. I’ve recently taken up tutoring work for work experience, and I hope to start tutoring A-level students online soon. That may look very pretentious and arrogant, but for context my teachers said I didn’t even need to turn up to A-level class when I formally “begin” the A-level next year.
Question:
Anyway, I am not a veteran teacher by any means. After the summer, when I’m tutoring for money or helping my friends, inevitably the courses will get to derivative calculus.
But when I was studying introductory calculus, I remember being extremely aggravated by not being told the reasons for things. In particular, the thing that really got me and held back my understanding for at least a month or so was: “but aren’t these calculus mathematicians just constantly dividing by zero? How can anyone call this formal! It’s sloppy maths!!” And while that is obviously a false statement that only a naive student would make, I couldn’t get over it - until I did. I can’t remember exactly what helped me: the idea that works for me now is the picture of the derivative as a limit, and the epsilon-delta definition is my friend here, but importantly it will not be taught to my friends or tutees during their time at secondary school!
So, I am dreading the day that one of my observant friends or observant tutees asks: “but aren’t we just dividing by zero?” because I can’t give a formal response, and I have zero proper educational background so I don’t know what the intuitive response should be, that actually works for pupils. I could point them to 3 blue 1 brown, but he does step into formalism here and there which is excellent for the students' learning and general appreciation but it is potentially scary for students just being introduced to calculus. Does anyone have any ideas, experience-backed, for how to take a curious student past the “derivative paradox”? It stumped my learning for a long time and for an A-level student with hardly any time at all that could be disastrous, and I don’t have the heart to tell them to just bottle up their curiosity and plough on with the course - that isn’t maths, that’s oppressive textbook education!
As a concrete example (but I invite the answerer to not focus on solely this example):
The power law for integer powers (and for non-integers too, but the base A-level doesn't cover the Taylor expansion of the binomial theorem) is very easily proven like this (this is a proof I thought of myself a while ago, I hope it's correct!)
$$\begin{align}\lim_{\delta x\to0}\frac{(x+\delta x)^n-x^n}{\delta x}&=\lim_{\delta x\to0}\frac{\sum_{k=0}^n{n\choose k}(\delta x)^k\cdot x^{n-k}-x^n}{\delta x}\\&=\lim_{\delta x\to0}\frac{(x^n-x^n)+n\cdot\delta x\cdot x^{n-1}+o(\delta x^2)}{\delta x}\\&=nx^{n-1}+\lim_{\delta x\to0}o(\delta x)\\&=nx^{n-1}\end{align}$$
Now my younger self would have had several quibbles with this: "why is $(\delta x)^2$ "smaller" than $\delta x$? $0$ is just $0$, right? And why are we doing maths where there are zeros on both the numerator, and the denominator, that's undefined,... right? What do you mean by vanishingly small?"
And I am personally a great believer in trying to prove things for oneself when learning; whenever I hear a Wikipedia article or a YouTube video state some fact, I always try to prove it for myself before continuing with the article/video. This is very rewarding, as it hones my formalism and understanding, and I would just love to ask a friend or tutee to prove the power law of derivatives using what they know about the binomial expansion and walk them through it but still leave the burden of proof with them... but I don't think I can, for A-level, judging by what I've seen of the A-level textbooks and so on. There are just too many unanswered quibbles!
Any advice is appreciated here. I want the best for my friends/tutees, and unanswerable quibbles due to the necessary evils of standardised, institutional education could damage their motivation/confidence that maths actually works.
To be very very precise: my question is about how to specifically soothe a curious student quibbling about the derivative “paradox” in the same way that I once (detrimentally, with no one to guide me) did. I’m not interested in general ideas of whether we should or shouldn’t push rigour into introductory calculus.