Once you reach the concept of differential forms in physics, a lot of these seemingly "ad-hoc" rules suddenly make sense. The issue is that outside of theoretic

Once you reach the concept of differential forms in physics, a lot of these seemingly "ad-hoc" rules suddenly make sense. The issue is that outside of theoretical physics grad students, few people will ever encounter those, since they basically only allow you to re-learn Maxwell's equations on a more fundamental level. But then again a lot of this directly translates into other field theories, including QFT, so there's that. For undergraduates, this seemingly wrong notation is mostly just taught out of convenience and readability. For example, you can do separation of variables more formally in undergrad notation by simply applying the integral as an operator to both sides. In your example that would be Integral(1 dy/dx dx) = Integral(f(x) dx), which works exactly the same, but the mnemonic "separation" of variables is no longer obvious at all.