Let R be a prime ring of characteristic different from 2, C the extended centroid of R, and $delta$ a generalized derivations of R. If [[$delta(x)$, x], $delta(x)$] = 0 for all $x;{in};R$ then either R is commutative or $delta(x);=;ax$ for all $x;{in};R$ and some $a;{in};C$. We also obtain some related result in case R is a Banach algebra and $delta$ is either continuous or spectrally bounded.