The aim of this paper is to select from the huge family of implicit two-step multiderivative methods those which have better characteristics as regards the error (order and error constant), stability and easy of use (with most of the coefficients equal to zero). A few of these multiderivative methods that are A0A0-stable and A(α)A(α)-stable have been obtained. This feature makes them useful for solving stiff problems. The stability intervals or stability regions for the methods chosen are presented. Some numerical tests are included. The numerical test concerning the wave equation shows the good performance of the methods of lower order but A0A0-stable, in comparison with the poor performance of higher-order methods.