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Open any book on differential geometry and compare the treatment of differentiation with the needlessly index heavy treatment in any undergraduate calculus textbook.

The point is that we treat the differential of a real valued function as a function/vector/matrix for historical reasons. The simpler perspective that always works is that the differential of a function is the best linear approximation of the function at a given point. But for historical reasons most math textbooks restrict themselves to "first order functions" and avoid, e.g., functions returning functions.

This also leads to ridiculous notational problems when dealing with higher order functions, like integration and all kinds of integral transforms.