In this paper we study the theory of ordered differential fields (CDO); in other words, the theory obtained adding the order axioms for a field to "differential field" 's axioms. If we consider a model K of such theory and we "forget" order, we know that such model is embedded in its differential closure. In a such closure, we can consider the set of real fields. Such a set has maximal elements (with respect to inclusion). We call CDO* the theory of so obtained maximal elements, for all $K \in Mod$ (CDO). If we neglect derivation, models of CDO* are real closed and then ordered. So we can prove that CDO* is the model completion of CDO. We find the axioms of CDO*, too. Finally, we find a method for eliminating quantifiers (for CDO*) in the formulas containing only inequalities.