Stability of linear systems with uncertain bounded time-varying delays is studied under the assumption that the nominal delay values are not equal to zero. An input-output approach to stability of such systems is known to be based on the bound of the L2-norm of a certain integral operator. There exists a bound on this operator in two cases: in the case where the delay derivative is not greater than 1 and in the case without any constraints on the delay derivative. In the present note we fill the gap between the two cases by deriving a tight operator bound which is an increasing and continuous function of the delay derivative upper bound d >1. For d → 1 the new bound corresponds to the second case and improves the existing bound. As a result, delay-derivative-dependent frequency-domain and time-domain stability criteria are derived for systems with the delay derivative greater than 1.