The core ingredients of the quantized calculus, introduced by A. Connes, are a separable Hilbert space H, a unitary self-adjoint operator F on H and a C -algebra A represented on H such that for all a 2 A the commutator [F; a] is a compact operator on H. Then the quantized di erential of a 2 A is de ned to be the operator da = i[F; a]. I will talk about the characterizations of the Schatten properties of quantum derivatives on quantum tori Td by Sobolev or Besov spaces.