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Let R be a prime ring of characteristic different from 2, C its extended centroid, d a nonzero derivation of R, f (x1, . . . , xn) a multilinear polynomial over C, ρ a nonzero right ideal of R andm > 1 a fixed integer such that ([d(f(r1, . . . , rn)), f(r1, . . . , rn)])m = [d(f(r1, . . . , rn)), f(r1, . . . , rn)] for all r1, . . . , rn ∈ ρ. Then either [f(x1,..., xn), xn+1]xn+2 is an identity for ρ or d(ρ)ρ = 0. [ABSTRACT FROM AUTHOR]