  In this paper, we show that if the product $(D_1D_2, d_1d_2)$ of generalized
derivations $(D_1, d_1)$ and $(D_2, d_2)$ on an algebra $A$ is a generalized
derivation, then $d_1D_2$ and $d_2D_1$ map $A$ into $\hbox{rad}(A)$. Also, for
generalized derivations $(D_1, d_1)$ and $(D_2, d_2)$ on a prime ring with
characteristic different from two, we give necessary and sufficient conditions
under which $(D_1^2+D_1D_2,d_1^2+d_1d_2)$ is a generalized derivation as well.
