Products of characters and derived length II
  Let $G$ be a finite solvable group and $\chi, \psi \in \Irr(G)$ be complex characters of $G$. Let $\alpha$ be an irreducible constituent of the product $\chi \psi$. We show that the derived length of $\Ker(\alpha)/\Ker(\chi\psi)$ is bounded by a linear function on the number of distinct irreducible constituents of $\chi\psi$ 
Edith Adan-Bante