The authors investigate the cohomology of infinitesimal quantum groups (and algebras) associated to classical quantum groups (and algebras) at a root of unity. A main result expresses the Ext-groups between irreducible modules for the Frobenius kernel in terms of those for the full quantum group. Under the assumption that certain module categories for the quantum group have a Kazhdan-Lusztig theory (in the sense of Cline, Parshall, Scott, \dots), this permits explicit calculations of cohomology in terms of Kazhdan-Lusztig polynomials. The similar calculations for the pull-back of the diagonal subgroup and the Frobenius kernel are also obtained. Considering that the cohomology of a quantum group is essentially the cohomology of comodules over a coalgebra, they have included an Appendix which collects together some elementary facts concerning comodule cohomology of a coalgebra.