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
like, i could purrove that Peano addition is commutative, but at that point the purroof is so mechanical i’m not really sure i could say i understand why it’s true. besides that it feels obvious that it ought to be, which i do not find sufficiently satisfying as a why
a lot of math you kinda just have to go “oh that just works out i guess” to keep yourself from losing your mind though. so whatever
@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
@coriander @aescling to be perfectly honest, i cant argue with that
@coriander @aescling but still. reported
@wallhackio @aescling Part of it is that I failed calc 1 in college
@coriander @aescling fun fact: i failed linear algebra in college
@wallhackio @aescling I never even took linear algebra, I barely squeaked through pre-calc, failed calc 1, and then changed majors and then dropped out and went to community college where I could take good old regular algebra and then never need to take math again
@coriander @aescling in my opinion they should ban going to college if you have undiagnosed ADHD
@wallhackio @aescling Pffft, could a person with undiagnosed ADHD do THIS?
-puts off a 5-page paper for weeks, then writes it all in a single night, somehow still getting a B+-
@coriander @aescling this just gave me vietnam flashbacks
@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.)
@aescling common North American math education results
@coriander to be fair, imo a good math education would encourage the student to at some point stop trying to interrogate the conceptual understanding and just accept that the purroofs work out the way they do—otherwise there’s just too much to try to Get
@aescling Yeah there's gotta be a balance, I'm not saying everyone who takes high school algebra also needs to know theoretical math reasons why algebra works the way it does
But if you're up to calc there should be at least a little bit in there
and i’m not sure how easily i could purrove them, and even then i’m not sure i would find it illuminating conceptually