Let R be a prime ring of characteristic different from 2 and F a b-generalized derivation on R. Let U be Utumi quotient ring of R with extended centroid C and f (x 1 ,. .. , x n) be a multilinear polynomial over C which is not central valued on R. Suppose that d is a non zero derivation on R such that d([F(f (r)), f (r)]) ∈ C for all r = (r 1 ,. .. , r n) ∈ R n ; then one of the following holds: (1) there exist a ∈ U, λ ∈ C such that F(x) = ax + λx + xa for all x ∈ R and f (x 1 ,. .. , x n) 2 is central valued on R, (2) there exists λ ∈ C such that F(x) = λx for all x ∈ R