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I'd like you to elaborate a bit more on the ill-defined inifinitesimals, because that sounds a bit like the way I was taught calculus in 2006. But maybe it's just that I studied engineering and we got the "light" part and not too rigurous. Or maybe we were taught in a horrid way and didn't know better.

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They are well-defined, but the usual framework that explains them ("non-standard analysis") is extremely abstract.

If you are dealing with polynomials/algebraic geometry only then there is a very simple, totally rigorous approach using nilpotents. And there is a newer approach called "alpha calculus" that I know nothing about but here is an example: https://arxiv.org/pdf/0807.3477.pdf

They are well-defined now, but when Leibniz and Cauchy were working with them, they were not, and used carelessly could lead to errors. Their "solution" was to just pretend those errors didn't happen ("don't do that, then") instead of trying to come up with a consistent system of rules for infinitesimals.

Infinitesimals have a long and complicated history (e.g. 1 = 0.999...) and their lack of a formal definition until recent years has lead to erroneous arguments such as thinking that (dx/dy) (dy/dz) (dz/dx) equals 1:

https://en.wikipedia.org/wiki/Triple_product_rule

I think you must mean dz/dx in the last term.

Thanks, fixed.

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