It’s one of the “nifty” features of Calculus, and in particular the Leibniz notation (dy/dx) that you can start with ∆y/∆x, then let ∆x→0, work through some algebra to see what it leads to as ∆x→0 (the “limiting process”) to find your derivative, and lo! and behold! you can get results that “appear” to let you treat dy/dx as a fraction in which the “numerator” and “denominator” can be treated as separate quantities.
(Just don’t try to cancel the “d” in the numerator and denominator, reducing dy/dx to y/x :dubious: )
Don’t they teach ε - δ definition of limits, and ε - δ proofs in First Semester Calculus anymore? When I took Calc I (circa 1982, Larson & Hostetler 2e), the definition was taught and a few trivially simple ε - δ proofs were shown, and we had exercise problems to do some other trivially simple ε - δ proofs. It was thus implied that all of the more seriously useful Calculus theorems (like the Chain Rule or all of L’Hôpital’s Rules, etc.) could be formally proved using ε - δ methods, although we didn’t actually do that.