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Density functional theory calculations on solids consisting of covalently bonded layers held together by dispersive interactions are presented. Utilizing the kinetic energy density in addition to the density and its gradients gives the meta-generalized gradient approximation (MGGA) M06-L enough flexibility to treat correctly both the covalent and the dispersive interactions in layered solids, thus making it a significant step forward compared to the local density and generalized gradient approximations. We show how the MGGA can take advantage of the extra information in the kinetic energy density to discriminate between dispersive and covalent interactions and thereby prove that the performance of M06-L for dispersive interactions, as opposed to that for the local density approximation, is not based on an accidental cancellation of errors.