Generic derivations on o-minimal structures

Let T be a complete, model complete o-minimal theory extending the theory RCF of real closed ordered fields in some appropriate language L. We study derivations I on models T. We introduce the notion of a T-derivation: a derivation which is compatible with the L()-definable 1-functions on . We show that the theory of T-models with a T-derivation has a model completion TGI. The derivation in models (I)TGI behaves "generically", it is wildly discontinuous and its kernel is a dense elementary L-substructure of . If T = RCF, then TGI is the theory of closed ordered differential fields (CODFs) as introduced by Michael Singer. We are able to recover many of the known facts about CODF in our setting. Among other things, we show that TGI has T as its open core, that TGI is distal, and that TGI eliminates imaginaries. We also show that the theory of T-models with finitely many commuting T-derivations has a model completion.