One other point about the earlier computation. Besides d^2 being 0 if d is the exterior derivative, there's another difficulty. Differential forms are elements

One other point about the earlier computation. Besides d^2 being 0 if d is the exterior derivative, there's another difficulty. Differential forms are elements of an exterior algebra Λ(W) where W is (say) a real vector space. Addition and multiplication of forms are defined; division is not. So before you can write "1/dx" and have it mean "division by dx", you have to first define what you mean by division by a differential form. The definition of the operation precedes the use of the symbol. I'm not sure how you'd define division of differential forms - you're going to take two differential forms, divide, and get a differential form? -but by all means explain if you have something in mind. I still think it was a good thought to ask whether "d/dx" could be interpreted as "exterior derivative divided by dx".