This manual is a direct continuation of the book by N.Ya. Vilenkin and E.S. Kunitskaya "Mathematical analysis. Introduction to Analysis." It contains a presentation of the course of differential calculus and its applications to the study of functions. The significance of this material for the future teacher is determined primarily by the fact that the relevant issues under the new program are studied in high school. One of the tasks of the manual is to clarify the basic concepts of differential calculus necessary for school teaching, strict proof of statements that are only explained at school. In this regard, much attention in the manual is paid to the natural science and geometric origins of the introduced concepts; questions of differentiation technique play a subordinate role.