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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 , . . . ,doi:10.2298/fil1919251t fatcat:d2435raz7za4tiocvumujblwxa