Identities with Engel conditions on derivations

Let R be a semiprime ring with a derivation D. The focus is on the two identities with Engel condition on D : [x(m), D(x(n1)),..., D(x(ns))](s) = 0 for all x. R and [x(m), D(x)(n1),..., D(x)(ns)](s) = 0 for all x. R, where s, m, n(1),..., n(s) are fixed positive integers. Our results are natural generalizations of Posner's theorem on centralizing derivations, Herstein's theorem on derivations with power-central values and a recent result by A. Fosner, M. Fosner and Vukman.