These Four Calculus Rules are Actually All the Same.

[@karelvanderwalt3625]

The usual sleigh of hand abuse of notation: jumping between d/dx operator to df/dx infinitesimals and dividing

[@ritvikmath]

So I usually think just in terms of ratios. Rather than d/dx being an "operator", I think of the total quantity df/dx as meaning "if I change x by a small amount dx, what is the relative magnitude of that change for f, as measured by df".

[@oraz.]

The product rule looks a bit like the definition of linearity but maybe it's justca coincidence.

[@SlovakiaPanda]

thank you it is so fun and intersting !!!!!!!!!!!!!!

[@ritvikmath]

thanks!

[@tsunningwah3471]

but for the product rule 5:10 u didn't show why are you adding them

[@ritvikmath]

that's fair, I could have explained that more. The idea is that those two paths are two independent ways by which a small change in x induces a small change in the product. Given this, we can sum the changes together since we can think of them as acting "one after the other"

[@HAGARCIA]

Thank you, million dollars for you!! Que legal!

[@ritvikmath]

Haha, glad you liked it!

[@HAGARCIA]

Sure! You showed enough to integrate the derivative sum to have the integrate one. I could understand your operation. I've never imagined this way! 😀

[@theproofessayist8441]

Oh no - the thumbnail looks like a diagram - do they commute (category theory) - are we referring to square of oppositions?