In this paper we investigate the conditions for derivations under which the Singer-Wermer theorem is true for noncommutative Banach algebra A such that either [[D(x),xD(x)] ${in}$ rad(A) for all $x{in}$A or $D(x)^2$x+xD(x))$^2$${in}$rad(A) for all $x{in}$A, where rad(A) is the Jacobson radical of A, then $D(A){subseteq}$rad(A).