The mixed discriminant of n Laurent polynomials in n variables is the irreducible polynomial in the coefficients which vanishes whenever two of the roots coincide. The Cayley trick expresses the mixed discriminant as an A-discriminant. We show that the degree of the mixed discriminant is a piecewise linear function in the Plücker coordinates of a mixed Grassmannian. An explicit degree formula is given for the case of plane curves.Dedicated to the memory of our friend Mikael Passare 2010 Mathematics Subject Classification. 13P15, 14M25, 14T05, 52B20.