  We conjecture results about the moments of mixed derivatives of the Riemann
zeta function, evaluated at the non-trivial zeros of the Riemann zeta function.
We do this in two different ways, both giving us the same conjecture. In the
first, we find asymptotics for the moments of derivatives of the characteristic
polynomials of matrices in the Circular Unitary Ensemble. In the second, we
consider the hybrid model approach first proposed by Gonek, Hughes and Keating.
