On derivatives of link polynomials

The higher order link polynomials are a class of link invariants related to both Homfly polynomial and Vassiliev invariants. Here we study their partial derivatives. We prove that each partial derivative of an nth order Homfly polynomial is an (n + 1)th order Homfly polynomial. In particular, each dth partial derivative of the Homfly polynomial is a dth order Homfly polynomial. For d = 1, we show that these derivatives span all the first order Homfly polynomials. Similar constructions are made for other link polynomials. Questions on linear span and computational complexities are discussed.