Derived subgroup not is powering-invariant

Statement
It is possible to have a group $$G$$ such that the derived subgroup $$G'$$ is not a powering-invariant subgroup of $$G$$. Explicitly, this means that there exists a prime number $$p$$ such that $$G$$ is powered over $$p$$ but the derived subgroup $$G'$$ is not.

Related facts
Both the facts below are strictly weaker, but it is relatively easier to construct the groups that provide examples for these, hence they may be more useful in that regard:


 * Derived subgroup not is divisibility-closed
 * Derived subgroup not is local powering-invariant

Proof
Taking the free $$\{ p \}$$-powered group on a set of size two or more works.