But as far as I know, there is no corresponding dual analysis. The dual numbers do not form a field.

The issue primarily revolves around elementary functions, compositions of them, and polynomials.

Yeah, I was only trying to explain where the issue (assuming smooth=analytic, that you rightfully pointed out) might have come from

Ahh, gotcha. I'm trying to figure this out myself. I've always been intrigued by automatic differentiation but have never been satisfied by any article addressing its mathematical justification.

Neither have I ...