The Derivative

Abstract

This chapter introduces the multivariable derivative, defined as a linear mapping and characterized intrinsically by the property of closely approximating the original mapping near the point of approximation. The derivative properties are established using the Bachmann–Landau notat–on scheme, allowing a range of ideas from calculus to be handled uniformly. The chapter then computes the derivative in coordinates, and it studies the multivariable counterparts of higher-order derivatives from one-variable calculus. Finally it discusses optimization and directional derivatives.