Let $R$ be a prime ring, $I$ a nonzero ideal of $R$, $d$ a derivation of $R$ and $m,n$ fixed positive integers. (i) If ${\left(d[x,y]\right)}^m={[x,y]}_n$ for all $x,y\in I$, then $R$ is commutative. (ii) If $\mathrm{Char}R\ne 2$ and ${[d\left(x\right),d\left(y\right)]}_m={[x,y]}^n$ for all $x,y\in I$, then $R$ is commutative. Moreover, we also examine the case when $R$ is a semiprime ring.

Let $R$ be a $2$-torsion free prime ring. Suppose that $\theta ,\phi$ are automorphisms of $R$. In the present paper it is established that if $R$ admits a nonzero Jordan left $(\theta ,\theta )$-derivation, then $R$ is commutative. Further, as an application of this resul it is shown that every Jordan left $(\theta ,\theta )$-derivation on $R$ is a left $(\theta ,\theta )$-derivation on $R$. Finally, in case of an arbitrary prime ring it is proved that if $R$ admits a left $(\theta ,\phi )$-derivation which acts also as a homomorphism (resp. anti-homomorphism) on a nonzero ideal...