A detailed proof of the incorrectness of the foundations of the differential calculus is proposed. The correct methodological basis for the proof is the unity of formal logic and rational dialectics. The unity of formal logic and rational dialectics is the only correct criterion of truth. The proof leads to the following irrefutable statement: differential calculus represents an incorrect theory in mathematics and physics. The proof of this statement is based on the following irrefutable results: (1) the standard theory of infinitesimals and the theory of limits underlying the differential calculus are incorrect theories. The concepts of "infinitesimal quantity", "movement", "process of tendency", and "limit of tendency" are meaningless concepts in mathematics; (2) the concepts of "increment of argument" and "increment of function" as the starting point of the differential calculus are not defined correctly; (3) the definition of the derivative of a function is an incorrect because the following logical contradiction arises: the increment of the argument is both not equal to zero and equal to zero; (4) the differentials of the argument and the function - as infinitesimal quantities - do not take on numerical values. This means that the differentials of quantities have neither quantitative nor qualitative determinacy; (5) the definition of the total differential of a function of two (many) variables does not satisfy the formal-logical law of the lack (absence) of contradiction; (6) the theory of proportions completely refutes the theory of differential calculus. Thus, differential calculus does not satisfy the criterion of truth and is not correct scientific (mathematical) theory.