Differentiability

The concept of derivative is the main theme of differential calculus, one of the major discoveries in mathematics, and in science in general. Differentiation is the process of finding the best local linear approximation of a function. The idea of the derivative comes from the intuitive concepts of velocity or rate of change, which are thought of as instantaneous or infinitesimal versions of the basic difference quotient ( f (x) - f (x ₀))/(x - x ₀), where f is a real-valued function defined in a neighborhood of x0. A geometric way to describe the notion of derivative is the slope of the tangent line at some particular point on the graph of a function.This means that at least locally (that is, in a small neighborhood of any point), the graph of a smooth function may be approximated with a straight line. Our goal in this chapter is to carry out this analysis by making the intuitive approach mathematically rigorous. Besides the basic properties, the chapter includes many variations of the mean value theorem as well as its extensions involving higher derivatives.