Let R be a semiprime ring with center Z(R). A mapping F : R → R (not necessarily additive) is said to be a multiplicative (generalized)derivation if there exists a map f : R→ R (not necessarily a derivation nor an additive map) such that F (xy) = F (x)y + xf(y) holds for all x, y ∈ R. The objective of the present paper is to study the following identities: (i) F (x)F (y)± [x, y] ∈ Z(R), (ii) F (x)F (y)± x ◦ y ∈ Z(R), (iii) F ([x, y]) ± [x, y] ∈ Z(R), (iv) F (x ◦ y) ± (x ◦ y) ∈ Z(R), (v) F ([x, y]) ± [F (x), y] ∈ Z(R), (vi) F (x ◦ y) ± (F (x) ◦ y) ∈ Z(R), (vii) [F (x), y] ± [G(y), x] ∈ Z(R), (viii) F ([x, y]) ± [F (x), F (y)] = 0, (ix) F (x ◦ y) ± (F (x) ◦ F (y)) = 0, (x) F (xy) ± [x, y] ∈ Z(R) and (xi) F (xy)±x◦y ∈ Z(R) for all x, y in some appropriate subset of R, where G : R → R is a multiplicative (generalized)-derivation associated with the map g : R→ R.