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Like one poster above said, df is a differential form, they live in the cotangent space. But cotangent space on R is isomorphic to R itself, so you can pullback that division operation to define it in the world of dx, dy and dz, then chain rule is literally cancelling out a fraction.
The bad news is you have to define derivative first to make these work. The badder news is if you go Rn, space of differentials is no longer isomorphic to R.
