Introduction to Mathematical Analysis I - Second Edition

DEFINITION AND BASIC PROPERTIES OF THE DERIVA- TIVE THE MEAN VALUE THEOREM SOME APPLICATIONS OF THE MEAN VALUE THEOREM L’HOSPITAL’S RULE TAYLOR’S THEOREM CONVEX FUNCTIONS AND DERIVATIVES NONDIFFERENTIABLE CONVEX FUNCTIONS AND SUBD- IFFERENTIALS 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. 4.1 DEFINITION AND BASIC PROPERTIES OF THE DERIVATIVE Let G be an open subset of R and consider a function f : G → R . For every a ∈ G , the function φ a ( x ) = f ( x ) − f ( a ) x − a is defined on G \ { a } . Since G is an open set, a is a limit point of G \ { a } (see Example 2.6.6 ) . Thus, it is possible to discuss the limit lim x → a φ a ( x ) = lim x → a f ( x ) − f ( a ) x − a . Definition 4.1.1 Let G be an open subset of R and let a ∈ G . We say that the function f defined on G is differentiable at a if the limit lim x → a f ( x ) − f ( a ) x − a exists (as a real number). In this case, the limit is called the derivative of f at a denoted by f 0 ( a ) , and f is said to be differentiable at a . Thus, if f is differentiable at a , then f 0 ( a ) = lim x → a f ( x ) − f ( a ) x − a . We say that f is differentiable on G if f is differentiable at every point a ∈ G . In this case, the function f 0 : G → R is called the derivative of f on G .