I find $dy/dx$ misleading because it treats $x$ and $y$ as similar objects.
When you use this notation, you lose the important point that $y$ is a function of $x$; instead you end up looking at $x$ and $y$ as related quantities.
I think it is important for calculus students to get the idea that differentiation is an operation that takes one function and produces a new function. In that way, it is fundamentally different from addition (or unary negation) of numbers (which is not the same thing as addition of functions).
Note that I am a lot more interested in (theoretical) computer science than (any form of) physics - this may bias my point of view.