People have mentioned the use of differential notation in differential equations. It's confusing. In teaching calculus we still use both the prime/Newton notat

People have mentioned the use of differential notation in differential equations. It's confusing. In teaching calculus we still use both the prime/Newton notation (y', y", and so on) and the fraction/Leibniz notation ("dy/dx", etc.) for derivatives. They're both useful for writing and for visualizing how things work, and both are used outside math (e.g. in physics and engineering). "dy/dx" was originally thought of as the quotient of two infinitesimal quantities --- arbitrarily small but (not necessarily) zero. Limits became accepted as the foundation for calculus, and infinitesimals were regarded as heuristic or non-rigorous. But if you're careful, "you can get the right answers", and Abraham Robinson's discovery of nonstandard analysis (around 1950 - 1960) showed that infinitesimals could be put on a rigorous basis. Besides Robinson's "Non-standard Analysis" and Jerome Keisler's (now free) calculus book someone else mentioned (and the supplement which provides a lot of the theory), people who are interested in this can check out "Infinitesimal Calculus" by James Henle and Eugene Kleinberg (https://store.doverpublications.com/0486428869.html https://store.doverpublications.com/0486428869.html - a short introduction) or "Applied Nonstandard Analysis" by Martin Davis (https://store.doverpublications.com/0486442292.html https://store.doverpublications.com/0486442292.html - it starts with the logic and set theory and like Keisler's supplement is fairly theoretical). Why aren't infinitesimals used in teaching calculus? After all, we can now make them rigorous, and the notation fits well with differential forms, which are used in differential geometry. Keisler's book (which I think came out in the 1970s) was an otherwise traditional calc book which made the attempt, but there is just too much inertia behind the limits approach. Some people were bothered also by the fact that infinitesimals are definitely non-constructive; the Bulletin of the American Math Society oddly had Errett Bishop, a noted constructivist, review Keisler's book when it came out, and the review was not very positive. (I don't know why they picked him as a reviewer.) But there's still a problem - you're defining the derivative as a limit of difference quotients, but it sure looks like a fraction. What to do? Calc books often fudge this in the following way. They define the derivative f'(x) (note the prime notation) as the limit of [f(x + h) - f(x)]/h as h goes to 0. Then "dx" is regarded as synonomous with "Δx", and both are considered to be simply increments/changes in x --- no infinitesimals. (So, e.g., dx could be 0.3.) Then you define dy to be f'(x) dx., so of course dy/dx = f'(x). In this interpretation, "dy" and "dx" are increments in y and x as you move along the tangent line to the curve. Rather than refer to these as "differentials", calc books will often put this in a section called "linear approximation", which usefully discusses using the tangent line to a curve to approximate the curve. I think it's good to bring up this history in calc courses - I always tried to. Math is rigorous and proceeds according to logic (that's the aspiration, anyway), but the way math develops and is taught is a social thing.