We study the nonlinear supercurrent response of unconventional superconductors to an applied magnetic field. We calculate numerically the superconducting penetration depth and the magnetization component transverse to the applied magnetic field, at finite temperature and in arbitrary field, in the Meissner state. In the d-wave pairing state we find that both quantities exhibit nonlinear effects, due to the presence of nodes in the order parameter. We relate the results to various experimental situations and show how one can verify whether an observed (T,H) is a signature of a particular pairing state. For an admixture of s-wave and d-wave superconducting states, we find that the transverse magnetization is suppressed, but that the s-wave component effect on the penetration depth may be overlooked in sufficiently large magnetic fields. We also consider dirty d-wave superconductors and discuss how these quantities, calculated as a function of temperature and field, are altered in this case.