I will present the following result:

Let (K, v, d) be a differential valued field with archimedean value group G = v(K -{0}) and residue field kv = R included in K.

The following two statements are equivalent:

(i) The derivation d is functional.

(ii) There exists a differential analytic morphism (K, v, d) ->

(R[[X^G]], ord, ?), where ? is some monomial derivation on the

generalised series field R[[X^G]].

A functional derivation carries an abstract version of properties that hold in the case of functions (continuity, L'Hospital's rule), for instance germs in a Hardy field. The notion of monomial derivation is a generalisation of the usual derivation for formal power series.