The purpose of this paper is to prove the following results: Let A be a noncommutative semisimple Banach algebra. (1) Suppose that a linear derivation D : A $ o A$ is such that [D(x),x]x=0 holds for all $x in A$. Then we have D=0. (2) Suppose that a linear derivation $D:A o A$ is such that $D(x)x^2 + x^2D(x)=0$ holds for all $x in A$. Then we have C=0.