Is it true that the derived series of $G/G^{(n-1)}$ is of length $n-1$, where $n$ is the length of the derived series of $G$?

Question

Consider a group $G$ and its derived series $$G=G^{(0)}\trianglerighteq G^{(1)} \trianglerighteq \ldots \trianglerighteq G^{(n-1)} \trianglerighteq G^{(n)}=\{1\}\,.$$ Is the derived series of $G/G^{(n-1)}$ is of length $n-1$? At first, I thought that it was clear because I thought that the derived series of $G/G^{(n-1)}$ was $$G/G^{(n-1)} \trianglerighteq G^{(1)}/G^{(n-1)}\trianglerighteq G^{(2)}/G^{(n-1)}\trianglerighteq \ldots \trianglerighteq G^{(n-1)}/G^{(n-1)}=\{e\}$$ But I'm not so sure if this is correct.