  We report constraints on the parameters of Yukawa-type corrections to
Newtonian gravity from measurements of the gradient of the Casimir force in the
configuration of an Au-coated sphere above a Si plate covered with corrugations
of trapezoidal shape. For this purpose, the exact expression for the gradient
of Yukawa force in the experimental configuration is derived and compared with
that obtained using the proximity force approximation. The reported constraints
are of almost the same strength as those found previously from several
different experiments on the Casimir force and extend over a wide interaction
range from 30 to 1260\,nm. It is discussed how to make them stronger by
replacing the material of the plate.
