Let $1<k$ and $m,k\in \mathbb{Z}^+$. In this manuscript, we analyse the action of (semi)-prime rings satisfying certain differential identities on some suitable subset of rings. To be more specific, we discuss the behaviour of the semiprime ring $\mathcal{R}$ satisfying the differential identities $([d([s,t]_m),[s,t]_m])^k=[d([s,t]_m),[s,t]_m]$ for every $s,t\in \mathcal{R}$.