Derived subgroup not is purely definable

History
This is based on as yet unpublished result of Bestvina and Feighn (referred to here).

Statement
The derived subgroup of a group need not be a purely definable subgroup.

More specifically, the derived subgroup of the free group of rank two is not a purely definable subgroup.

Related facts

 * Center is purely definable
 * Finite upper central series members are purely definable