Let R be a prime ring, I a nonzero ideal of R and n a fixed positive integer. If R admits a generalized derivation F associated with a derivation d such that c for all x, $y{in}I$. Then either R is commutative or n = 1, d = 0 and F is the identity map on R. Moreover in case R is a semiprime ring and $(F([x,;y]))^n=[x,;y]$ for all x, $y{in}R$, then either R is commutative or n = 1, $d(R){subseteq}Z(R)$, R contains a non-zero central ideal and for all $x{in}R$.