ABSTRACT
Of course the derivative is a significant idea from calculus, and it is important that authors establish a rigorous and precise understanding of the concept. The authors’ aim is to develop the derivative as a precise analytic tool, and also to get a rigorous proof of the Fundamental Theorem of Calculus. This chapter begins authors' discussion of the derivative by establishing some basic properties and relating the notion of derivative to continuity. All differentiable functions are continuous: differentiability is a stronger property than continuity. Since the subject of differential calculus is concerned with learning uses of the derivative, it concentrates on functions which are differentiable. Therefore it came as a shock when Karl Weierstrass produced a continuous function that is not differentiable at any point. The proof of Weierstrass's theorem is long, but the idea is simple: the function f is built by piling oscillations on top of oscillations. The Mean Value Theorem is a powerful analytic tool.
