In topology, we may consider a topological space $X$. A subset $S$ of $X$ has its "derived set" $S'$, the set of all limit points of $S$. On the other hand, in algebra, we have a group $G$, and the derived subgroup $G' = [G, G]$ is the set of all finite products of commutators of $G$.
Moreover, we sometimes consider the chains of derived sets $$A' \supset A'' \supset A''' \supset \cdots $$ and derived subgroups $$G' \supset G'' \supset G''' \supset \cdots $$ That is about all I know that is common between the two structures. Is there a deeper notion of "derived" going on here, between topological spaces and groups, and perhaps other structures, or is it just two superficially similar operations that happen to share the same name?