PREVIEW VERSION—NOT FINAL
3 DIFFERENTIATION
ifferential calculus is the study of the derivative, and differentiation is the process of D computing derivatives. What is a derivative? There are three equally important an- swers: Aderivative is a rate of change, it is the slope of a tangent line, and (more formally), it is the limit of a difference quotient, as we will explain shortly. In this chapter, we explore all three facets of the derivative and develop the basic rules of differentiation. When you master these techniques, you will possess one of the most useful and flexible tools that mathematics has to offer.
3.1 Definition of the Derivative We begin with two questions: What is the precise definition of a tangent line?And how can we compute its slope? To answer these questions, let’s return to the relationship between Calculus is the foundation for all of our tangent and secant lines first mentioned in Section 2.1. understanding of motion, including the The secant line through distinct points P = (a, f (a)) and Q = (x, f (x)) on the graph aerodynamic principles that made supersonic of a function f (x) has slope [Figure 1(A)] flight possible. − f = f (x) f (a) x x − a
where f = f (x) − f (a) and x = x − a f (x) − f (a) The expression is called the difference quotient. REMINDER A secant line is any line x − a through two points on a curve or graph. y y Q = (x, f(x))
P = (a, f(a)) Δf = f(x) − f(a) Tangent P = (a, f(a))