Let $R$ be a 2-torsion free prime ring and let $\sigma ,\tau$ be automorphisms of $R$. For any $x,y\in R$, set ${[x,y]}_{\sigma ,\tau }=x\sigma \left(y\right)-\tau \left(y\right)x$. Suppose that $d$ is a $(\sigma ,\tau )$-derivation defined on $R$. In the present paper it is shown that $\left(i\right)$ if $R$ satisfies ${[d\left(x\right),x]}_{\sigma ,\tau }=0$, then either $d=0$ or $R$ is commutative $\left(ii\right)$ if $I$ is a nonzero ideal of $R$ such that $\left[d\right(x),d(y\left)\right]=0$, for all $x,y\in I$, and $d$ commutes with both $\sigma$ and $\tau$, then either $d=0$ or $R$ is commutative. $\left(iii\right)$ if $I$ is a nonzero ideal of $R$ such that $d\left(xy\right)=d\left(yx\right)$, for all $x,y\in I$, and $d$ commutes with $\tau$, then $R$ is commutative. Finally a related result has been obtain for $(\sigma ,\tau )$-derivation....