ABSTRACT
In elementary calculus, the main mathematical tool for studying the behavior of functions was the derivative. In this chapter, the authors define the derivative of a function between Euclidean spaces, develop some of its properties, and show how to compute it. They provide some examples of the derivative of a function between Euclidean spaces. The authors show that a differentiable function has a unique derivative. They examine the chain rule for determining the derivative of the composite of two differentiable functions and also provide some examples to show how it is used. The authors also show that if the partial derivatives of a function exist and are continuous, then the function is differentiable. They discuss some preliminary results which are of the independent interest.
