A Note on Derivations in Rings and Banach Algebras

Let $r$ be a prime ring with $u$ the utumi quotient ring and $q$ be the martindale quotient ring of $r$, respectively. let $d$ be a derivation of $r$ and $m,n$ be fixed positive integers. in this paper, we study the case when one of the following holds:$(i)$~ $d(x)circ_n d(y)$=$xcirc _m y$ $(ii)$~$d(x)circ_m d(y)$=$(d(xcirc y))^n$ 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 apply our result to the continuous derivations on banach algebras.