We investigate the joint moments of derivatives of characteristic polynomials over the unitary symplectic group Sp(2N) and the orthogonal ensembles SO(2N) and O−(2N) . We prove asymptotic formulae for the joint moments of the n 1th and n 2th derivatives of the characteristic polynomials for all three matrix ensembles. Our results give two explicit formulae for each of the leading order coefficients, one in terms of determinants of hypergeometric functions and the other as combinatorial sums over partitions. We use our results to put forward conjectures on the joint moments of derivatives of L-functions with symplectic and orthogonal symmetry.