Let $R$ be a prime ring, $F$ be a generalized derivation associated with a derivation $d$ of $R$ and $m,n$ be the fixed positive integers. In this paper we study the case when one of the following holds: (i) $F(x)\circ_m F(y)=(x\circ y)^n$, (ii) $F(x)\circ_m d(y)=d(x\circ y)^n$ for all $x,y$ in some appropriate subset of $R$. We also examine the case where $R$ is a semiprime ring. Finally, as an application we obtain some range inclusion results of continuous or spectrally bounded generalized derivations on non-commutative Banach algebras.