Beware of the Train

Although actually, my comment wasn't entirely facetious. You say that someone learning conventional calculus can happily manipulate "dx" on its own without the term having a meaning. That doesn't go for me - I didn't like using it at all, and so didn't (except in integration, where it does have a specified meaning) -- until I came to a context where it was in fact defined to mean something. And I think I'll largely be strongly disuading my students from using it. There's never any need, anyway.