ABSTRACT
This chapter provides the basic result of Taylor's theorem. The basic result gives conditions under which a function can be approximated near a point by a polynomial. As in the case of real-valued functions of one variable, higher derivatives play an important role in the study of functions between Euclidean spaces. The higher derivatives can again be considered as linear transformations; however, our emphasis will be on the higher order partial derivatives. The general form of Taylor's theorem is notationally quite complicated. The chapter deals with an example showing how the chain rule is used to compute higher order partial derivatives of composite functions. Although in principle this is a straightforward application of the chain rule, in practice it can be quite complicated.
