Differential algebras without differentials: an easy C++ implementation

It is shown that automated differentiation can be motivated and explained rather plainly without any reference to infinitesimals or differentials whatsoever. The proposed approach clears up the confusion connected either with employing the notion of differential or with the connection to *R and its implied machinery of ultrafilters. An almost trivial implementation of this method using C++ is described. In the proposed approach, differentiation becomes an arithmetic, rather than analytic, procedure, thereby rendering it appropriate for automated computation