Let R be a prime ring with characteristics not 2 and ${sigma},;{ au},;{alpha},;{eta}$ be auto-morphisms of R. Suppose that $d_1$ is a (${sigma},;{ au}$)-derivation and $d_2$ is a (${alpha},;{eta}$)-derivation on R such that $d_{2}{alpha};=;{alpha}d_2,;d_2{eta};=;{eta}d_2$. In this note it is shown that; (1) If $d_1d_2$(R) = 0 then $d_1$ = 0 or $d_2$ = 0. (2) If [$d_1(R),d_2(R)$] = 0 then R is commutative. (3) If($d_1(R),d_2(R)$) = 0 then R is commutative. (4) If $[d_1(R),d_2(R)]_{sigma, au}$ = 0 then R is commutative.