one of those weird things about math is that i “know basic calculus” in that i know the fundamental differential rules, but i feel like i don’t really “know calculus” in that i could not even begin to explain why they work
@aescling if you use the limit definition of the derivative and approximate to lowest order in dx you can straightforwardly make a physicist's derivation of all the rules like the product rule and the division rule and many differention rules like the power rule or derivatives of sines and cosines
@wallhackio @aescling Yes-and I personally think the multi-variable forms are much easier to understand, and justify all the weird results and derivative tricks you have to memorize in single-variable calc.
I wish more people would teach it first, though there's probably a fair bit of "suspension of disbelief"-style "just go with this" needed at the start since the real-world applications aren't immediately obvious if you start with multiple variables.
@wallhackio @aescling (Also I'm pretty sure I've said this before, probably to the same people, sorry, but the multi-variable chain rule makes all of the "derivative of {addition, multiplication, division, exponentiation, logarithm, etc}" stuff make a lot more sense, at least if you can understand those as multivariable functions themselves. (Which is easy for me, but I also program computers and grok RPN, so I can't vouch for it being intuitive to most people.)