Let R be a prime ring with characteristic different from two and let $ heta;varphi,;sigma,; au$ be the automorphisms of R. Let d : $R; ightarrow;R$ be a nonzero ($ heta,;varphi$)-derivation. We prove the following results: (i) if $a;in;R;and;[d(R),;a]_{ hetaosigma,;varphio au};=;0,;then;sigma(a);+; au(a);in;Z$, the center of R, (ii) if $d([R,a]_{sigma,; au);=;0,;then;sigma(a);+; au(a);in;Z$,, (iii) if $[ad(x),;x]_{sigma,; au};=;0;for;all;x;in;RE, then a = 0 or R is commutative.