Differential calculus, which arose more than three hundred years ago in the works of Newton and Leibniz, opened a new era in the development of science. It served as the basis for the creation of modern mathematics and found numerous applications in natural science and technology. This brochure introduces the basic concepts of differential calculus: limit, derivative, continuity of function, and describes the application of these concepts in mechanics, biology, sociology, and other fields. The reader will also learn about how scientists’ ideas about differential calculus have changed over the past three centuries. The text of the brochure is an additional processing of the recording of the lecture "Extremums of functions of one variable", read by the author on February 24, 2001 at the Small Mekhmat of Moscow State University for schoolchildren of grades 9-11 . The brochure is intended for a wide range of readers interested in mathematics: high school students, junior students, teachers ...