Definition 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 defined on an open interval I = (a,b), where a,b∈Randa<b. 4.1 Definition and Basic Properties of the Derivative Let f : I →Rbe a function, where I is an open interval. For every x0 ∈I, the function φx0(x) = f (x)−f (x0) x−x0 is defined onI \ {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 lim x→x0 φx0(x) = lim x→x0 f (x)−f (x0) x−x0 . Definition 4.1.1 Let I be an open interval inRand let x0 ∈I. We say that the function f defined on I is differentiable at x0 if the limit lim x→x0 f (x)−f (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 ′(x0) = lim x→x0 f (x)−f (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 →Ris called the derivative of f on I.
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