Abstract: Use the r-th classic modulus of smoothness ωr（f，t） to study the relation of the derivatives between Bernstein-Kantorovich operators and the smoothness of the function it approximates,and obtain the equivalent theorem between the derivatives of Bernstein-Kantorovich operators and the r-th classic modulus of smoot
hness ωr（f，t）.

Key words: Bernstein-Kantorovich operators, modulus of smoothness, derivatives