  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 $\delta$ on models $\mathcal{M}\models T$. We introduce the notion
of a $T$-derivation: a derivation which is compatible with the
$L(\emptyset)$-definable $\mathcal{C}^1$-functions on $\mathcal{M}$. We show
that the theory of $T$-models with a $T$-derivation has a model completion
$T^\delta_G$. The derivation in models $(\mathcal{M},\delta)\models T^\delta_G$
behaves "generically," it is wildly discontinuous and its kernel is a dense
elementary $L$-substructure of $\mathcal{M}$. If $T =$ RCF, then $T^\delta_G$
is the theory of closed ordered differential fields (CODF) 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 $T^\delta_G$ has $T$ as its open
core, that $T^\delta_G$ is distal, and that $T^\delta_G$ eliminates
imaginaries. We also show that the theory of $T$-models with finitely many
commuting $T$-derivations has a model completion.
