"dx^2 is negligible" can be alternatively (rigorously) obtained in standard analysis/calculus in the following sequence: 0) Intermediate value theorem (continu

"dx^2 is negligible" can be alternatively (rigorously) obtained in standard analysis/calculus in the following sequence: 0) Intermediate value theorem (continuous function with image between b and c must be such that there is x such that f(x)=d for d between b and c. This can be first proved where d=0 -- it's the method of bisection by nested intervals, really -- and then generalized) 0.5) The actual definition of derivatives known to everyone. 1) Rolle's theorem, i.e. the Intermediate Value Theorem on first derivatives (if something goes up and down then it must have a critical pont). 2) Mean Value Theorem (Rolle's theorem but rotated) 3) Taylor's theorem.