Derived lengths and character degrees

Let $G$ be a finite solvable group. Assume that the degree graph of $G$ has exactly two connected components that do not contain $1$. Suppose that one of these connected components contains the subset $\{ a_{1}, \dots , a_{n} \}$, where $a_{i}$ and $a_{j}$ are coprime when $i \not = j$. Then the derived length of $G$ is less than or equal to $|\operatorname{cd}(G)|-n+1$.