Notice the difference between the thinking of a physicist vs a mathematician, Newton vs Leibniz.

Newton's trick of making his curve analysis into a mechanical problem, implies absolute time (by parametrizing of the curve using another independent variable). This was a good approximation. It wasn't until Einstein, that it was realized that time can't always be used as an independent variable. It depends on the relative velocity of a linearly moving frame of reference, and also on your gravitational field.

Leibniz' trick of geometry, was time agnostic. These are not equivalent POV in general, though in some cases they produce the same results. Newton got priority eventually, but he tied calculus to mechanics. Leibniz notation and geometric view is what is taught today (except in physics classes).