高等数学06VIP免费

目录上页下页返回结束二、反函数的求导法则三、复合函数求导法则四、初等函数的求导问题一、四则运算求导法则函数的求导法则第二章目录上页下页返回结束解决求导问题的思路:(构造性定义)求导法则其它基本初等函数求导公式0xcosx1)(C)sin(x)ln(x证明中利用了两个重要极限初等函数求导问题本节内容目录上页下页返回结束一、四则运算求导法则定理1.的和、差、积、商(除分母为0的点外)都在点x可导,且下面分三部分加以证明,并同时给出相应的推论和例题.可导都在点及函数xxvvxuu)()()()(xvxu及)()(])()([)1(xvxuxvxu)()()()(])()([)2(xvxuxvxuxvxu)()()()()()()()3(2xvxvxuxvxuxvxu)0)((xv目录上页下页返回结束此法则可推广到任意有限项的情形.证:设则vuvu)()1(故结论成立.例如,,)()()(xvxuxfhxfhxfxfh)()(lim)(0hxvxuhxvhxuh)]()([)]()([lim0hxuhxuh)()(lim0hxvhxvh)()(lim0)()(xvxuwvuwvu)(目录上页下页返回结束(2)vuvuvu)(证:设,)()()(xvxuxf则有hxfhxfxfh)()(lim)(0hxvxuhxvhxuh)()()()(lim0故结论成立.)()()()(xvxuxvxuhhxuh)(lim0)(xu)(hxvhxv)()(xu)(hxv推论:)()1uC)()2wvuuCwvuwvuwvu)log()3xaaxlnlnaxln1(C为常数)目录上页下页返回结束)()(lim0xvhxvh)()()()()()(xvhxvhxvxuxvhxuh)()(xvxu(3)2vvuvuvu证:设)(xf则有hxfhxfxfh)()(lim)(0hhlim0,)()(xvxu)()(hxvhxu)()(xvxuhhxu)()(xu)(xvhhxv)()(xu)(xv故结论成立.)()()()()(2xvxvxuxvxu推论:2vvCvC(C为常数)目录上页下页返回结束例1.解:xsin41(21)1sin,)1sincos4(3xxxyy)(xx)1sincos4(213xxx23(xx)1xy1cos4)1sin43(1cos21sin2727)1sincos4(3xx)1sincos4(3xx目录上页下页返回结束)(cscxxsin1x2sin)(sinxx2sin例2.求证证:xxxcossin)(tanx2cosxxcos)(sin)(cossinxxx2cosx2cosx2sinx2secxcosxxcotcsc类似可证:,csc)(cot2xx.tansec)(secxxx目录上页下页返回结束)(xf二、反函数的求导法则定理2.y的某邻域内单调可导,证:在x处给增量由反函数的单调性知且由反函数的连续性知因此()(),xyyfx设为的反函数在()0y且ddxy或,0x)()(xfxxfy,0xyyx,00yx时必有xyxfx0lim)(lim0yyxyxdd1()y11()y11目录上页下页返回结束例3.求反三角函数的导数.解:1)设则,)2π,2π(y)(sinyycos1y2sin11类似可求得xxarcsin2πarccos利用0cosy,则目录上页下页返回结束)arcsin(x)arccos(x)arctan(x)cotarc(xaaaxxln)(xxe)e(小结:目录上页下页返回结束在点x可导,lim0xxyxyx0limdd三、复合函数求导法则定理3.在点可导复合函数且)()(ddxgufxy在点x可导,证:)(ufy在点u可导,故)(lim0ufuyuuuufy)(（当时)故有)()(xgufuy)(uf)0()(xxuxuufxy目录上页下页返回结束例4.求下列导数:解:(1))(e)(lnxx)ln(xx1x)(e)(lnxxxx)ln(xxxx)1ln(x(2)222arcsin22xaxyaxya练习：已知，求关键:搞清复合函数结构,由外向内逐层求导.目录上页下页返回结束例5.设112xxy求解:)cos(e1x))sin(e(xxe)tan(eexx22(1)21xx112x解:目录上页下页返回结束四、初等函数的求导问题1.常数和基本初等函数的导数(P95))(C0)(x1x)(sinxxcos)(cosxxsin)(tanxx2sec)(cotxx2csc)(secxxxtansec)(cscxxxcotcsc)(xaaaxln...