It's funny that most intro calculus courses will make it a point to remind you that "dy/dx" isn't a fraction, then when they get to integration & diffeqs they want you to forget that and start manipulating them as such. I think most intro courses would be better off skipping everything on convergence tests (which feel really arbitrary anyway until you understand more of complex analysis) and instead use that time better explaining differentials (and maybe a peek into differential forms)
Related: "Putting Differentials Back into Calculus " at https://bridge.math.oregonstate.edu/papers/differentials.pdf
I remember reading Einstein's Relativity and having to translate the notation into what I was learning in Calculus class.
There's other goofy stuff people do with df/dx, right? Like in a u-substitution you literally do "algebra" with it.
It's funny that most intro calculus courses will make it a point to remind you that "dy/dx" isn't a fraction, then when they get to integration & diffeqs they want you to forget that and start manipulating them as such. I think most intro courses would be better off skipping everything on convergence tests (which feel really arbitrary anyway until you understand more of complex analysis) and instead use that time better explaining differentials (and maybe a peek into differential forms)
If you thought that was goofy, check out "Umbral calculus" https://en.wikipedia.org/wiki/Umbral_calculus
well.. no, not exactly. If u = u(x) then du = u'(x) dx holds rigorously, and then you can substitute du/u' = dx in an integral.
I'm thinking more along the lines of knocking a '2x' out of an integral from d/dx of like 2x^2.