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大学文科数学第三章教案.pdf

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1、文艺复兴的火炬驱散了欧洲中世纪的漫漫黑暗，15 世纪之后的欧洲，资本主义逐渐，出现的大量实际问题，给数学提出了前所未有的亟待解决的新课题，其中三类问题导致了微分学的产生：（1）求变速运动的瞬时速度（2）求曲线上一点的切线（3）求极大值和极小值1.1 抽象导数概念的两个现实原型原型 I 求变速直线运动的速度设一质点 M 从点 O开始做变速直线运动，经过T 秒到达 P 点，求该质点在00,tT 时刻的瞬时速度.以 O为原点，沿质点运动的方向建立数轴-s轴，用s表示质点的运动的路程，显然路程s是时间 t 的函数，记作(),0,sf ttT，现求00,tT时刻的瞬时速度00()vv t.如果质点做匀速

2、直线运动，那么按照公式=路程速度时间，便可以求出0v，但是现在要求质点做变速直线运动的速度，则在整个时间间隔0,T 内不能应用上边的公式求0t时刻的速度0v，下面我们分三步来解决这一问题.(1)给0t一个增量t，时间从0t变到10ttt，质点 M 从点0M运动到点1M，路程有了增量1000sf tf tf ttft(2)当t 很小时，速度来不及有较大的变化，可以把质点在t间隔内的运动看似匀速运动，这实质上是把变速运动近似的转化为匀速运动，下面求t内的平均速度00fttftsvtt(3)当t 越来越小，平均速度就越来越接近于0t时刻的瞬时速度0v，即000000limlimlimtttfttf

3、tsvvtt原型 II 求曲线切线的斜率在初等数学中，我们知道曲线)(xfy上的两点000(,)Mxy和,Mx y 的连线为曲线的割线，当点 M 沿着曲线无限的趋近于0M时，其极限位置就是曲线在点0M处的切线，如何求曲线在0M处的切线的斜率呢？我们分三步来解决：(1)求增量给0 x一个增量x，自变量由0 x变到xx0，曲线上纵坐标的相应增量为y=00()()f xxf x.(2)求增量比曲线)(xfy上的点从000(,)Mxy变到00,Mxx yy 时，当x 很小时，此时曲线上的纵坐标来不及有很大的变化，这时候割线的斜率近似的等于切线的斜率，此时割线0M M的斜率为xxfxxfxy)()(00

4、(3)取极限当0 x时，点00,Mxx yy 沿着曲线无限的接近000(,)Mxy，割线0M M的斜率的极限就是切线的斜率，即0000()()tanlimlimxxf xxf xyxx其中2，是切线与x轴正向之间的夹角.1.2 导数概念定义设函数xfy在点0 x的某邻域内有定义，当自变量x有一个增量x 时，相应函数值的增量为y=00()()f xxf x，若极限000limxfxxfxx存在，则称函数f在点0 x可导，并称该极限为函数f在点0 x处的导数，记为0 xf，0 xxy，0 xxdxdy，0 xxdxdf等.若上述极限不存在，则称f在点0 x不可导.导数是函数增量y与自变量增量x

5、之比xy的极限，这个增量比称为函数关于自变量的平均变化率，而导数0 xf=000limxxxfxfxx是函数在点0 x处的变化速度，称为函数f在点0 x处的瞬时变化率.导数的力学意义就是变速直线运动物体的瞬时速度导数的几何意义就是曲线的切线斜率文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H

6、5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3

7、Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI

8、4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B

9、3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG

10、3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B

11、9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编

12、码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5例 1 求函数2()f xx在点2x处的导数解：给2x一个增量x，0(2)(2)(2)limxfxffx20(2)4limxxx20444limxxxx4如果函数f在区间(,)a b内每一点都可导，则称f为区间(,)a b上的可导函数。此时对每一个(,)xa b，都有f的一个导数xf与之对应，记作xf，y，dxdy，dxdf等.即xxfxxfxfx0lim这就是说：函数xf在点0 x的导数0 xf是曲线xfy在点0 x处的函数值例 2 求函数1yx在点1x处的导数解：0()()()limxf xxf xfxx011l

13、imxxxxx01limxx xx21x(1)1f例 3 求函数x的导数解：0()()()limxf xxf xfxx0limxxxxx0limxxxxxx12x综上面的例题，幂函数x的导数1xx例 4 求常数函数Cy的导数.解：(1)求增量：因为Cy，即不论x取什么值，y的值总等于C，所以0y；文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G

14、9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7

15、ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T

16、6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文

17、档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH

18、7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10

19、Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10

20、HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5(2)算比值：xy0；(3)取极限：00limlim00 xxxyy.即常数函数的导数等于零.例 5 求函数xysin的导数.解（1）求增量：xxxxfxxfysin)sin()()(，由和差化积公式有：2)(sin2)(cos2xxxxxxy（2）算比值：22sin)2cos(2sin)2cos(2xxxxxxxxxy.（3）取极限：22sin)2cos(limlimdd00 xxxxxyxyxx00sin2lim cos()limcos22xxxxxxx即(si

21、n)cosxx，用类似的方法，可求得(cos)sinxx我们同样可以利用导数定义去证明对数函数exxaalog1log，特别地xx1ln1.5 函数的可导性与连续性之间的关系定理 2 若函数xf在x处可导，则函数xf在x处连续.1.6 高阶导数的概念函数xf的变化率是用它的导数()fx来表示的，而导数()fx也是x的函数，那么函数()fx的变化率也应该用它的导数()fx来表示，我们把它称为函数xf的二阶导数，记作xf，22d ydx文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9

22、R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码

23、:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5

24、Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z

25、10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4

26、X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3

27、Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3

28、K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5二阶导数的力学意义就是运动物体的加速度设函数xf存在1n阶导数，并且1n阶导数可导，那么11nnyfx 的导数称为函数xf的n阶导数记为xfn,xyn,nndxyd例设yx，1,10yxy设sinyx，cosyx，sinyx课堂练习：78P 3.（1）10.(3)总结：1、学习导数的基本定义及导数的几何意义2、掌握函数导数的求法

29、作业：78P 3.（2）2.1 求导法则1.函数的和、差、积、商的求导法则(1)设函数）（xuu与）（xvv在点x处可导,则函数 u xv x（）（）也在点x处可导,且有以下法则:u xv xu xv x（）（）（）（）例1 已知3sinln 2yxx解：33sinln 2sin(ln 2)yxxxx23cosxx(2)设函数）（xuu与）（xvv在点x处可导,则函数）（）（xvxu也在点x处可导,且有以下法则:)()()()()()(xvxuxvxuxvxu证明：令）（）（xvxuy,(1)求函数y的增量:给x以增量x,相应地函数）（xu,）（xv各有增量u 与v,从而y有增量,vxuxxu

30、vxvxxvxuxxvxuxxuxvxuxxvxxuy）（）（）（）（）（）（）（）（）（）（）（）（文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B

31、9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编

32、码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H

33、5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3

34、Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI

35、4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B

36、3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R52)算比值:xuxuxxvxuxy）（）（,(3)取极限:由于）（

37、xu与）（xv均在 x 处可导,所以）（），（xvxvxuxuxx00limlim.又,函数）（xv在x处可导,就必在x处连续,因此)()(lim0 xvxxvx,从而根据和与乘积的极限运算法则有.limlimlimlim0000）（）（）（）（）（）（xvxuxvxuxvxuxxvxuxyxxxx这就是说，）（）（xvxuy也在 x 处可导且有）（）（）（）（）（）（xvxuxvxuxvxu.特别的，当,()uCCvCv例2 已知2ln2cosyxxxx解：22ln2cosln2cosyxxxxxxxx2112 ln2(cos-sin)2xxxxxxxxcos2 ln2sinxxxxxxx(

38、3)设函数）（xuu与）（xvv在点x处可导,则函数）（）（）（0 xvxvxu也在点x处可导,且有以下法则:）（）（）（）（）（）（）（xxxuxxuxxu2)0)(xv特别的，当21,uvuvv例 3 已知xytan，求y解：2sinsincossincostancoscosxxxxxyxxx（）（）（）（）文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z1

39、0 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X

40、4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z

41、7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K

42、7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R

43、5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:

44、CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y

45、10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R522222cossin1sec,coscosxxxxx即xx2sectan）（同理可得xx2csccot）（例 4 已知xysec,求y解：21cosseccoscosxyxxx（）（）=2sinsec tancosxxxx同理可得xxxcotcsccsc）（2.复合函数的求导法则设函数)(xfy是由函数)(ufy和)(xu复合而成的函数，并且设函数)(xu在点 x 处可导，)(ufy在对应的点)(xu处可导，则有复合函数的求导法则：xuuyxydd

46、dddd也可表示为 ()()()fxf ux复合函数的导数等于函数对于中间变量的导数乘以中间变量对于自变量的导数.例 5 求xysin的导数解：函数xysin可以看作由函数uysin与xu复合而成因此xxxuxuy2cos21cos)()(sin例 6 ln cos()yx，求y解：函数是由ln,cos,yu uv vx复合而成，则lncosyuvx11sin2vux11sincos2xxx例 7 lnyx，求y解：ln,0lnln(),0 x xyxxx文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3

47、Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3

48、K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9

49、R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码

50、:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5Y10Z3Z10 HI4X4G9B3Z7 ZG3K7T6B9R5文档编码:CH7H5

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