The concept of "immediate mappings" on ultrametric spaces is a generalization of the notion of immediate extensions of valued fields. A main theorem giving a criterion for the subjectivity of immediate mappings provides a uniform tool to prove all sorts of generalized Hensel's Lemmas, among them differential Hensel's Lemmas for both D-fields in the sense of Scanlon and differential valuations in the sense of Rosenlicht.

After presenting a quick introduction to immediate mappings and the main theorem, I will show how these differential Hensel's Lemmas are derived. In the case of D-fields, the Hensel's Lemma we obtain is satisfactory. But for the Rosenlicht case, it is very restricted; I will discuss the problems that occur in this case.