An apparatus of partial and external derived numbers is proposed, which makes it possible to investigate the behavior of a function of several variables, without requiring its differentiability, but using only information about partial derived numbers, reducing restrictions on the degree of smoothness of the functions studied. The use of the apparatus of external derived numbers makes it possible to reduce the restrictions on the degree of smoothness of manifolds when studying the question of the integrability of the field of hyperplanes. The proposed method can be used to obtain the necessary or sufficient conditions for the stability of solutions of systems of differential equations. Using the apparatus of partial and external derived numbers, it can be shown that the investigation of the stability of solutions of a system of differential equations can be reduced to an investigation of the solvability of a system of equations of a special form.