Defnition and Basic Properties of the Derivative The Mean Value Theorem Some Applications of the Mean Value Theorem L’Hôpital’s Rule Taylor’s Theorem 4. DIFFERENTIATION In this chapter, we discuss basic properties of the derivative of a function and several major theorems, including the Mean Value Theorem and L’Hôpital’s Rule. Throughout this chapter, we consider functions defned on an open interval I =(a,b), where a,b ∈ R and a < b. 4.1 Defnition and Basic Properties of the Derivative Let f : I → R be a function, where I is an open interval. For every x0 ∈ I, the function f (x) − f (x0) (x)= φx0 x− x0 is defned on I \{x0}. Since I is an open interval, x0 is a limit point of I \{x0} (see Exercise 3.1.4). Thus, it is possible to discuss the limit f (x) − f (x0) lim φx0 (x)= lim . x→x0 x→x0 x − x0 Defnition 4.1.1 Let I be an open interval in R and let x0 ∈ I. We say that the function f defned on I is differentiable at x0 if the limit f (x) − f (x0) lim x→x0 x − x0 exists (as a real number). In this case, the limit is called the derivative of f at x0 denoted by f ′ (x0), and f is said to be differentiable at x0. Thus, if f is differentiable at x0, then f (x) − f (x0) f ′ (x0)= lim . x→x0 x − x0 We say that f is differentiable on I if f is differentiable at every point x0 ∈ I. In this case, the function f ′ : I → R is called the derivative of f on I.
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