ABSTRACT
This chapter develops the theory of differentiation based on the definition of Cauchy, with special emphasis on the mean value theorem and consequences thereof. It presents the standard results concerning derivatives of functions obtained by means of algebraic operations and composition. In the examples and exercises the chapter derives the derivatives of some of the basic algebraic and trigonometric functions. However, throughout the chapter the author assumes that the reader is already familiar with standard techniques of differentiation and some of its applications. As a consequence the chapter concentrates on the mathematical concepts of the derivative, emphasizing many of its more subtle properties. It proves the mean value theorem and give several consequences of this important result. Even though the proof itself is elementary, the theorem is one of the most useful results of analysis. Its importance is based on the fact that it allows reader to relate the values of a function to values of its derivative.
