Separant of an arbitrary polynomial

Let $f$ be a unitary polynomial over $F$. Previously, the concept of a separant of a polynomial $f$ was defined for the case where f has no multiple roots. The notion of a separant turned out to be very useful for generalizations of Hensel's lemma. We propose a generalization of this concept to the case where a polynomial may have multiple roots. This allows us to extend Hensel's lemma to this case as well.