In this paper, we prove that biorthogonal ensembles on the real line with a specific derivative structure admit an explicit correlation kernel of double contour integral form. We will demonstrate that this expression is a valuable starting point for asymptotic analysis and that our class of biorthogonal ensembles admits a large variety of limit kernels, by proving that two new classes of limit kernels can occur. The first type is a deformation of the hard edge Bessel kernel which arises in polynomial ensembles describing the eigenvalues of the sum of two random matrices, while the second type arises for Muttalib-Borodin type deformations of polynomial ensembles.