It Don't Work That Way

Every calculus student's favorite party trick: treating derivatives like fractions and just canceling stuff out. The student here is doing some algebraic gymnastics by "canceling" the dx terms like they're simple fractions, which... technically gives you the right answer but makes every differential equations professor age 10 years instantly.

Here's the thing: the chain rule actually DOES give you dy/dt = (dy/dx)(dx/dt), so the student isn't wrong. They're just using freshman algebra logic on calculus notation, which is like using a hammer to perform surgery—it might work, but you're gonna give someone a heart attack watching you do it. The professor's horrified reaction is because while the notation looks like fractions, they're actually limits of ratios, and treating them as algebraic fractions is mathematically sketchy even if it works out.

It's the mathematical equivalent of getting the right answer for the wrong reasons, and professors can smell that from a mile away.