Solvability of fixed length is direct product-closed

Version in terms of fixed class bound
Suppose $$G_i, i \in I$$ is a collection of groups indexed by an indexing set $$I$$. Suppose there is a positive integer $$\ell$$ such that each $$G_i$$ is a solvable group of derived length at most $$\ell$$.

Then, the external direct product of the $$G_i$$s is also a solvable group of derived length at most $$\ell$$.

Version in terms of maximum class
Suppose $$G_i, i \in I$$ is a collection of groups indexed by an indexing set $$I$$. If all the $$G_i$$s are solvable groups and there is a common finite bound on their derived length values, then the external direct product of the $$G_i$$s is also a solvable group and its derived length is the maximum of the derived length values of all the $$G_i$$s.

In particular, for two solvable groups $$G_1$$ and $$G_2$$ of derived lengths $$\ell_1,\ell_2$$ respectively, the derived length of $$G_1 \times G_2$$ equals $$\max \{ \ell_1, \ell_2 \}$$.

Related facts

 * Nilpotency of fixed class is direct product-closed
 * Derived series is direct product-closed