Is `dy/dx` ever distinct from `d/dx y`? Generally I've treated `df(x)/dx` as just another way to write the operator `d/dx` applied to `f(x)`, which I don't think of as a fraction.

dy/dt = dy/dx * dx/dt

That's just the chain rule, right?

d/dt y(t) = ( d/dx y(x) )( d/dt x(t) )

What series of operations can lead you to end up with dy/dx though? Like Integral{x^2){dx} is how you're introducing dx into a formula, but it's not like integration just works over division, so I'm a bit stuck on the construction phase

y = x^2

dy = 2x dx

dy/dx = 2x

Chain rule makes this not completely trivial formulation.