In a recent paper we have shown that Conway's field No of surreal numbers admits a differential structure extending the one on transseries and compatible with infinite sums and the exponential function. This makes No into a Liouville-closed H-fields; in particular every element has an anti-derivative. Moreover, the work of Aschenbrenner, van den Dries and van der Hoven shows that any Hardy field embeds in the differential field No. So far so good, but is there a "better" derivation? To make this precise, we explore the possibility of defining a composition of surreal numbers satisfying the chain rule. We give a positive answer if we restrict ourselves to the (proper class) subfield generated by the reals and the ordinal number omega using the field operations, the exponential function, and infinite sums. Finally, we give some arguments "against" our derivation, showing that it cannot be compatible with a composition on the whole class of surreals.

This is joint work with Vincenzo Mantova.