Differential algebraic dependence and Novikov dependence

We define an analogue of the Fox derivatives for differential polynomial algebras and give a criterion for differential algebraic dependence of a finite system of elements. In particular, we prove that differential algebraic dependence of a finite set of elements of a differential polynomial algebra over a constructive differential field $k$ of characteristic zero is algorithmically recognizable. Using a representation of free Novikov algebras by differential polynomials we also give a criterion of Novikov dependence of a finite system of elements of free Novikov algebras.