Why is dy/dx = (dy/du)(du/dx) an insufficient proof of the Chain Rule? In almost all Calculus textbooks, dy/dx = (dy/du)(du/dx) is given as a good rationale for the Chain Rule, but they always say that a more rigorous proof is needed, since du can't equal 0. But du isn't 0, is it? It approaches 0, and both du's should approach 0 at the same "rate," since they're the same quantity. What's the problem with canceling them, then? Is it just because it doesn't use the definition of the derivative (with the epsilon-delta definition of a limit)?