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The ω-and p-meson tensor couplings to nucleons in a derivative scalar coupling model for finite nuclei are investigated. The influences of the tensor couplings on the binding energies per nucleon, the root-mean-square charge radii, spin-orbit splittings and single particle energies are discussed. The obtained results show that the spin-orbit splittings for finite nuclei are more sensitive to the ω-meson tensor coupling.