足以保证f（x）/g（x） 同理，若是-∞ P< f(x) (a 高阶导数 定义 如果在一个区间上有导函数f＇，而ｆ＇自身又是可微的，把ｆ 的导函数记做ｆ＇＇，就叫做ｆ的二阶导数。照这样继续下去就得到 （３）（ｎ） ｆ，ｆ＇，ｆ＇＇，ｆ，．．．，ｆ， （ｎ） 这许多函数，其中每一个是前一个的导函数。ｆ叫做f的n阶导函数。 （ｎ）（） 为了要ｆ（x）在x点存在fn-1（x）必须在x点的某个邻域例存在（当 （） x是定义f的区间的端点时，fn-1（x）必须在它有意义的那个单侧邻域例存在）， （）（） 而且fn-1必须在x点可微。因为在fn-1必须在x的邻域例存在，那么，f（n-2）又必须在x的邻域里可微。 文献原文附录： Walter.Rudin. principles of mathematical analysis[M].北京：机械工业出版社，2004 5 DIFFERNTIATION In this chapter we shall(except in the final section)confine our attention to real functions dfine on intervals or segements.This is not just a matter of convenience,since genuine differences appear when we pass from real functions to vector-valued ones.Differentiation of functions defined on Rk will be discussed in Chap 9. THE DERIVATIVE OF A REAL FUNCTION 5.1 Definition Let f be defined (and real-valued) on [a,b]. Fof any x∈[a,b] from the quotient (1) and define (2) provided this limit exists in accordance with Definition 4.1 We thius associante with the function f a function f’ whose domain is the set of points x at which the limit (2) exists; f’ is called the derivative of f. If f’ is defined at a point x, we say that f is differentiable at x. If f’ is defined at every point of a set E∈[a,b],we say that f is differentiable on E. It is possible to consider right-hand and left-hand derivative,respectively.We shall not,however,discuss one-side derivatives in any detail. If f is defined on a segment(a,b) and if a 5.2 Theorem Let be defined on [a,b].If it is differentiable at a point x ∈[a,b],then f is continuous at x. Proof As t→x,we have,by Theorem 4.4 The converse of this theorem is not true. It is easy to construct continuous functions which fail to be differentiable at isolated points.In Chap 6 we shall even become acquainted with a function which is continuous on the whole line without being differentiable at any point! 5.3 Theorem Suppose f and g are defined on [a,b] and are differentiable at a point x∈[a,b].Then f+g,fg,and f/g are differentiable at x,and (a) (f+g)’(x)=f’(x)+g’(x); (b) (fg)’(x)=f’(x)g(x)+f(x)g’(x); 2 (c) (f/g)’(x)=[g(x)f’(x)-g’(x)f(x)]/g(x) In (c),we assume of course that g(x)≠0. Proof (a) is clear,by Theorem 4.4.Let h=fg.Then If we divide this by t-x and note that f(t)→f(x) as t→x(Theorem5.2),(b) follows. Next, let h=f/g. Then Letting t→x,and applying Teorems 4.4 and 5.2, we obtain(c). 5.4 Examples The derivative of any constant is clearly zero.If f is defined by n f(x)=x,then f’(x)=1. Repeated application of (b) and (c) then shows that x is n-1 differentiable,and that its derivative is nx,for any integer n (if n<0, we have to restrict ourselves to x ≠0).Thus every polynomial is differentiable,and so is every rational function,except at the points where the denominator is zero. The following theorem is known as the “chain rule”for differentiation.It deals with differentiation of composite functions and is probably the most important theorem about derivateves.Ws shall meet more general versions of it in Chap 9. 5.5 Theorem Suppose f is continuous on [a,b],f’(x) exists at some point x∈[a,b],g is defined on an interval I which contains the range of f,and g is differentiable at the point f(x).If h(t)=g(f(t)) (a≤t≤b), then h is differentiable at x,and (3) h’(x)=g’(f(x))f’(x). Proof Let y=f(x).By the definition of the derivative,we have (4) (5) where t∈[a,b],s∈I,and u(t)→0 as t→x,v(s)→0 as s→y.Let s=f(t). Using first (5) and then (4),we botain or,if t≠x, (6) Letting t→x,we see that s→y,by the continuity of f,so that the right side of (6) tends to g’(y)f’(x),which gives (3). 5.6 Examples (a) Let f be defined by (7) Taking for granted that the derivative of sinx is cos x (we shall discuss the trigonometric functions in Chap 8),we can apply Theorems 5.3 and 5.5 whenever x≠0,and obtain (8) At x=0,these theorems do not apply any longer,since 1/x is not denfined there,and we appeal directly to the definition:for t≠0, As t→o,this does not tend to any limit,so that f’(0) does not exist. (b) Let f be defined by (9) As above,we obtain (10) At x=0,we appeal to the definition,and obtain Letting t→0,we see that (11) Thus f is differentiable at all points x,but f’ is not a continuous function,since cos(1/x) in (10) does not tend to a limit as x→0. MEAN VALUE THEOREMS 5.7 Definition Let f be a real function defined on a metric space X.We say That f has a local maximum at a point p∈X if there exists ?>0 such that f(q)≤f(p) for all q∈X with d(p,q)< ?. Local minima are defined likewise. Our next theorem is the basis of many applications of differentiation. 5.8 Theorem Let f be defined on [a,b];if f has a local maximum at a point x∈(a,b),and if f’(x) exisets,then f’(x)=0. The analogous statement for local minima is of course also ture. Proof choose ? in accordance with Definition 5.7,so that If x- ? Letting t→x,we see that f’(x)≥0. If x Which shows that f’(x)≤0.Hence f’(x)=0. 5.9 Theorem If f and g are continuous real functions on [a,b] which are differentiable in (a,b), then there is a point x∈(a,b)at which Note that differentiability is not required at the endpointds. Proof Put Then h is continuous on [a,b],h is differentiable in (a,b),and (12) To prove the theorem,we have to show that h’(x)=0 for some x∈(a,b). If h is constant,this holds for every x∈(a,b).If h(t)>h(a) for some t∈(a,b),let x be a point on [a,b] at which h attains its maximum(Theorem 4.16). By(12),x∈(a,b),and Theorem 5.8 shows that h’(x)=0.If h(t) This theorem is often called a generalized mean value theorem;the Following special case is usually referred to as “the”mean value theorem: 5.10 Theorem If f is a real continuous function on [a,b] which is differetiable in (a,b),then there is a point x∈(a,b) at which Proof Take g(x)=x in Theorem 5.9. 5.11 Theorem Suppose f is differentiable in (a,b). (a) If f’(x)≥0 for all x∈(a,b),then f is monotonically increasing. (b) If f’(x)=0 for all x∈(a,b),then f is constant. (c) If f’(x)≤0 for all x∈(a,b),then f is monotonically decreasing. Proof All conclusions can be read off from the equation Which is valid,for each pair of numbers x1,x2in (a,b),for some x between x1 and x2. THE CONTINUITY OF DERIVATIVES We have already seen [Example 5.6(b)]that a function f my have a derivative f’which exisets at every point,but is discontinuous at some point. Howere,not every function is a derivative.In particular,derivatives Which exist at every point of an interval have one important property in common with functions which are continuous on an interval:Intermediate values are assumed(compare Theorem 4.23).The pecise statement follows. 5.12 Theorem Suppose f is a real differentiable function on [a,b] and Suppose f’(a) A similar result holds of course if f’(a)>f’(b). Proof Put g(t)=f(t)- ?t.Then g’(a)<0,so that g(t1) Corollary If f is a differentiable on [a,b],then f’cannot have any simple discontinuities on [a,b]. But f’ my very well have discontinuities of the second kind. L’HOSPITAL’S RULE The following theorem is frequently useful in evaluation of limits. 5.13 Theorem Suppose f and g are real and differentiable in (a,b),and g’(x)≠0 for all x∈(a,b),where -∞≤a≤b≤+∞.Suppose (13) If (14) Or if (15) then ), (16) The analogous statement is of course also true if x→b,or if g(x) →-∞ in (15). Let us note that we now use the limit concept in the extend sense of Definition 4.33
《数学分析原理》
