Nadeem Rehman - Department of mathematics, Aligarh Muslim University, Aligarh, India.
Shuliang Huang - School of Mathematics and Finance, Chuzhou University, Chuzhou, Anhui Province, P.R.China
Mohd Raza - Department of Mathematics, Faculty of Science and Arts-Rabigh, King Abdulaziz University, Jeddah, Kingdom of Saudi Arabia.

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)=x\circ _m y (ii)~d(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 apply our result to the continuous derivations on Banach algebras.

Prime and semiprime rings, Derivations, Martindale ring of quotients, Banach algebras, Radical