Our object of study is the simplest nontrivial sweeping process,
specified by the movement of an interval of fixed size within
one-dimensional space (the real numbers). We address the existence
and form of weak derivatives of the associated nonsmooth solution
operator in function space, and show in particular that its
directional derivative is the unique solution of a certain
evolution variational inequality. Then we use this variational
inequality to derive optimality conditions for a related optimal
control problem in form of a strong stationarity system, that is,
a system which is equivalent to the basic necessary first order
condition.