Understanding a proof of: If $N\unlhd G$ s.t. $N$ and $G/N$ are solvable, then $G$ is solvable.
Maybe you should proceed by a small example: let's start with the first derivative, namely $(G/N)'=G'N/N$. How do you prove this? Well just write it out: $$(G/N)'=\langle [\overline{x},\overline{y}]: \overline{x},\overline{y} \in G/N \rangle \\=\langle [x,y]N: x,y \in G \rangle \\=\langle [x,y]: x,y \in G \rangle N \\=G'N/N.$$ And $G'N/N=\overline{1}$ if and only if $G'N=N$ if and only if $G' \subseteq N$. Now this should help you for the higher derivatives.