Abstract Higher order link polynomials were defined by combining ingredients from link polynomials and Vassiliev invariants. It has been proved that each nth partial derivative of the Homfly polynomials is an nth order Homfly polynomial. This naturally raises two questions: Question 1. Are these partial derivatives linearly independent? Question 2. Do they span the space of higher order link polynomials? In this paper, we give an affirmative answer to Question 1. As a by-product, we determine all the higher order Conway polynomials.