Summary: Let \(R\) be a prime ring with its Utumi ring of quotient \(U\), \(C=Z(U)\), the extended centroid of \(R,G\) a generalized derivation of \(R\) and \(\lambda\) a nonzero ideal of \(R\). Suppose that there exists \(0\ne b\in R\) such that \(b([x,y]t[G([x,y]),[x,y]][x,y]s)m=0\) or \(b((x\circ y)^t[G(x\circ y),(x\circ y)](x\circ y)^s)^m\) for all \(x,y\in\lambda\), where \(t\ge 1\), \(s\ge 0\), \(m\ge 1\) are fixed integers. Then either \(R\) satisfies the standard identity \(s_4(x_1,x_2,x_3,x_4)\) in four variables \(x_1,x_2,x_3,x_4\) and \(G(x)=qx+xq+\alpha x\) for some \(q\in U\) and \(\alpha\in C\) or \(G(x)=\alpha x\) for all \(x\in R\) with \(\alpha\in C\).