Differentiation of real valued functions

Abstract

We now begin the differential calculus for real valued functions of several variables. The first step is to define the notions of directional derivative and partial derivative. Then the concept of differentiable function is introduced, by linear approximation to the increments of a function. Taylor’s formula with remainder is obtained for functions of class C^(q); such functions have continuous partial derivatives of orders 1, 2,..., q. It is then applied to problems of relative extrema and to the characterization of convex functions of class C⁽²⁾.