Approximation of differentiation operator in the space $L_2$ on semiaxis

We establish an upper bound for the error of the best approximation of the first order differentiation operator by linear bounded operators on the set of twice differentiable functions in the space $L_2$ on the half-line. This upper bound is close to a known lower bound and improves the previously known upper bound due to E. E. Berdysheva. We use a specific operator that is introduced and studied in the paper.