反三角函数求导公式的证明.pdf
名师推荐精心整理学习必备反三角函数求导公式的证明2.3 反函数的导数，复合函数的求导法则一、反函数的导数设)(yx是直接函数，)(xfy是它的反函数，假定)(yx在Iy内单调、可导，而且0)(y，则反函数)(xfy在间,)(|yxIyyxxI内也是单调、可导的，而且)(1)(yxf (1)证明：xIx，给x以增量x),0(xIxxx由)(xfy在Ix上的单调性可知0)()(xfxxfy于是yxxy1因直接函数)(yx在Iy上单调、可导，故它是连续的，且反函数)(xfy在Ix上也是连续的，当0 x时，必有0y)(11limlim00yyxxyyx即：)(1)(yxf【例 1】试证明下列基本导数公式().(arcsin)().()().(log)ln1112113122xxarctgxxaxax证 1、设yxsin为直接函数，xyarcsin是它的反函数函数yxsin在)2,2(yI上单调、可导，且xycos0因此，在)1,1(xI上，有学习资料总结-名师归纳欢迎下载-欢迎下载 名师归纳-第 1 页，共 5 页 -名师推荐精心整理学习必备yxcos1)arcsin(注意到，当)2,2(y时，0cos y，221sin1cosxyy因此，211)arcsin(xx证 2 设xtgy，)2,2(yI则yarctgx，Ix(,)tgyx在Iy上单调、可导且0cos12yx故2221111cos)(1)(xytgytgyarctgx证 3 axaaaayyxln1ln1)(1)log(类似地，我们可以证明下列导数公式：(arccos)()(ln)xxarcctgxxxx1111122二、复合函数的求导法则如果)(xu在点x0可导，而)(ufy在点)(00 xu可导，则复合函数)(xfy在点x0可导，且导数为)()(000 xufdxdyxx学习资料总结-名师归纳欢迎下载-欢迎下载 名师归纳-第 2 页，共 5 页 -文档编码:CJ10K3G4Y3M6 HF3R3H1Y1P3 ZJ9G4V2O10X2文档编码:CJ10K3G4Y3M6 HF3R3H1Y1P3 ZJ9G4V2O10X2文档编码:CJ10K3G4Y3M6 HF3R3H1Y1P3 ZJ9G4V2O10X2文档编码:CJ10K3G4Y3M6 HF3R3H1Y1P3 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xuxuufxyxxxuxuufxxx0000limlimlim)()()(00 xuf即)()(000 xufdxdyxx上述复合函数的求导法则可作更一般的叙述：若ux()在开区间Ix可导，yf u()在开区间Iu可导，且xIx时，对应的uIu，则复合函数)(xfy在Ix内可导，且dxdududydxdy (2)复合函数求导法则是一个非常重要的法则，特给出如下注记：学习资料总结-名师归纳欢迎下载-欢迎下载 名师归纳-第 3 页，共 5 页 -文档编码:CJ10K3G4Y3M6 HF3R3H1Y1P3 ZJ9G4V2O10X2文档编码:CJ10K3G4Y3M6 HF3R3H1Y1P3 ZJ9G4V2O10X2文档编码:CJ10K3G4Y3M6 HF3R3H1Y1P3 ZJ9G4V2O10X2文档编码:CJ10K3G4Y3M6 HF3R3H1Y1P3 ZJ9G4V2O10X2文档编码:CJ10K3G4Y3M6 HF3R3H1Y1P3 ZJ9G4V2O10X2文档编码:CJ10K3G4Y3M6 HF3R3H1Y1P3 ZJ9G4V2O10X2文档编码:CJ10K3G4Y3M6 HF3R3H1Y1P3 ZJ9G4V2O10X2文档编码:CJ10K3G4Y3M6 HF3R3H1Y1P3 ZJ9G4V2O10X2文档编码:CJ10K3G4Y3M6 HF3R3H1Y1P3 ZJ9G4V2O10X2文档编码:CJ10K3G4Y3M6 HF3R3H1Y1P3 ZJ9G4V2O10X2文档编码:CJ10K3G4Y3M6 HF3R3H1Y1P3 ZJ9G4V2O10X2文档编码:CJ10K3G4Y3M6 HF3R3H1Y1P3 ZJ9G4V2O10X2文档编码:CJ10K3G4Y3M6 HF3R3H1Y1P3 ZJ9G4V2O10X2文档编码:CJ10K3G4Y3M6 HF3R3H1Y1P3 ZJ9G4V2O10X2文档编码:CJ10K3G4Y3M6 HF3R3H1Y1P3 ZJ9G4V2O10X2文档编码:CJ10K3G4Y3M6 HF3R3H1Y1P3 ZJ9G4V2O10X2文档编码:CJ10K3G4Y3M6 HF3R3H1Y1P3 ZJ9G4V2O10X2文档编码:CJ10K3G4Y3M6 HF3R3H1Y1P3 ZJ9G4V2O10X2文档编码:CJ10K3G4Y3M6 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ZJ9G4V2O10X2文档编码:CJ10K3G4Y3M6 HF3R3H1Y1P3 ZJ9G4V2O10X2文档编码:CJ10K3G4Y3M6 HF3R3H1Y1P3 ZJ9G4V2O10X2文档编码:CJ10K3G4Y3M6 HF3R3H1Y1P3 ZJ9G4V2O10X2文档编码:CJ10K3G4Y3M6 HF3R3H1Y1P3 ZJ9G4V2O10X2文档编码:CJ10K3G4Y3M6 HF3R3H1Y1P3 ZJ9G4V2O10X2文档编码:CJ10K3G4Y3M6 HF3R3H1Y1P3 ZJ9G4V2O10X2名师推荐精心整理学习必备弄懂了锁链规则的实质之后，不难给出复合更多层函数的求导公式。【例 2】)(xfy，求dydx引入中间变量，设vx()，uv()，于是yf u()变量关系是yuvx，由锁链规则有：dydxdydududvdvdx(2)、用锁链规则求导的关键引入中间变量，将复合函数分解成基本初等函数。还应注意：求导完成后，应将引入的中间变量代换成原自变量。【例 3】求yxsin2的导数dydx。解：设ux2，则yusin，ux2，由锁链规则有：dydxdydududxuxux(sin)()(cos)cos2222【例 4】设ytgxln2，求dydx。由锁链规则有dxdvdvdududydxdy21cos112vu(基本初等函数求导)212cos1212xxtg(消中间变量)xsin1由上例，不难发现复合函数求导窍门学习资料总结-名师归纳欢迎下载-欢迎下载 名师归纳-第 4 页，共 5 页 -文档编码:CJ10K3G4Y3M6 HF3R3H1Y1P3 ZJ9G4V2O10X2文档编码:CJ10K3G4Y3M6 HF3R3H1Y1P3 ZJ9G4V2O10X2文档编码:CJ10K3G4Y3M6 HF3R3H1Y1P3 ZJ9G4V2O10X2文档编码:CJ10K3G4Y3M6 HF3R3H1Y1P3 ZJ9G4V2O10X2文档编码:CJ10K3G4Y3M6 HF3R3H1Y1P3 ZJ9G4V2O10X2文档编码:CJ10K3G4Y3M6 HF3R3H1Y1P3 ZJ9G4V2O10X2文档编码:CJ10K3G4Y3M6 HF3R3H1Y1P3 ZJ9G4V2O10X2文档编码:CJ10K3G4Y3M6 HF3R3H1Y1P3 ZJ9G4V2O10X2文档编码:CJ10K3G4Y3M6 HF3R3H1Y1P3 ZJ9G4V2O10X2文档编码:CJ10K3G4Y3M6 HF3R3H1Y1P3 ZJ9G4V2O10X2文档编码:CJ10K3G4Y3M6 HF3R3H1Y1P3 ZJ9G4V2O10X2文档编码:CJ10K3G4Y3M6 HF3R3H1Y1P3 ZJ9G4V2O10X2文档编码:CJ10K3G4Y3M6 HF3R3H1Y1P3 ZJ9G4V2O10X2文档编码:CJ10K3G4Y3M6 HF3R3H1Y1P3 ZJ9G4V2O10X2文档编码:CJ10K3G4Y3M6 HF3R3H1Y1P3 ZJ9G4V2O10X2文档编码:CJ10K3G4Y3M6 HF3R3H1Y1P3 ZJ9G4V2O10X2文档编码:CJ10K3G4Y3M6 HF3R3H1Y1P3 ZJ9G4V2O10X2文档编码:CJ10K3G4Y3M6 HF3R3H1Y1P3 ZJ9G4V2O10X2文档编码:CJ10K3G4Y3M6 HF3R3H1Y1P3 ZJ9G4V2O10X2文档编码:CJ10K3G4Y3M6 HF3R3H1Y1P3 ZJ9G4V2O10X2文档编码:CJ10K3G4Y3M6 HF3R3H1Y1P3 ZJ9G4V2O10X2文档编码:CJ10K3G4Y3M6 HF3R3H1Y1P3 ZJ9G4V2O10X2文档编码:CJ10K3G4Y3M6 HF3R3H1Y1P3 ZJ9G4V2O10X2文档编码:CJ10K3G4Y3M6 HF3R3H1Y1P3 ZJ9G4V2O10X2文档编码:CJ10K3G4Y3M6 HF3R3H1Y1P3 ZJ9G4V2O10X2文档编码:CJ10K3G4Y3M6 HF3R3H1Y1P3 ZJ9G4V2O10X2文档编码:CJ10K3G4Y3M6 HF3R3H1Y1P3 ZJ9G4V2O10X2文档编码:CJ10K3G4Y3M6 HF3R3H1Y1P3 ZJ9G4V2O10X2文档编码:CJ10K3G4Y3M6 HF3R3H1Y1P3 ZJ9G4V2O10X2文档编码:CJ10K3G4Y3M6 HF3R3H1Y1P3 ZJ9G4V2O10X2文档编码:CJ10K3G4Y3M6 HF3R3H1Y1P3 ZJ9G4V2O10X2文档编码:CJ10K3G4Y3M6 HF3R3H1Y1P3 ZJ9G4V2O10X2文档编码:CJ10K3G4Y3M6 HF3R3H1Y1P3 ZJ9G4V2O10X2文档编码:CJ10K3G4Y3M6 HF3R3H1Y1P3 ZJ9G4V2O10X2文档编码:CJ10K3G4Y3M6 HF3R3H1Y1P3 ZJ9G4V2O10X2文档编码:CJ10K3G4Y3M6 HF3R3H1Y1P3 ZJ9G4V2O10X2文档编码:CJ10K3G4Y3M6 HF3R3H1Y1P3 ZJ9G4V2O10X2文档编码:CJ10K3G4Y3M6 HF3R3H1Y1P3 ZJ9G4V2O10X2文档编码:CJ10K3G4Y3M6 HF3R3H1Y1P3 ZJ9G4V2O10X2文档编码:CJ10K3G4Y3M6 HF3R3H1Y1P3 ZJ9G4V2O10X2文档编码:CJ10K3G4Y3M6 HF3R3H1Y1P3 ZJ9G4V2O10X2文档编码:CJ10K3G4Y3M6 HF3R3H1Y1P3 ZJ9G4V2O10X2文档编码:CJ10K3G4Y3M6 HF3R3H1Y1P3 ZJ9G4V2O10X2文档编码:CJ10K3G4Y3M6 HF3R3H1Y1P3 ZJ9G4V2O10X2文档编码:CJ10K3G4Y3M6 HF3R3H1Y1P3 ZJ9G4V2O10X2文档编码:CJ10K3G4Y3M6 HF3R3H1Y1P3 ZJ9G4V2O10X2文档编码:CJ10K3G4Y3M6 HF3R3H1Y1P3 ZJ9G4V2O10X2文档编码:CJ10K3G4Y3M6 HF3R3H1Y1P3 ZJ9G4V2O10X2名师推荐精心整理学习必备中间变量在求导过程中，只是起过渡作用，熟练之后，可不必引入，仅需“心中有链”。然后，对函数所有中间变量求导，直至求到自变量为止，最后诸导数相乘。请看下面的演示过程：)2(2cos121)2(21)2ln(2xxxtgxtgxtgxtgdxdyxxxtgxxxtgsin122cos21)(212cos12122【例 5】证明幂函数的导数公式1)(xx，(为实数)。证明：设yxexln1lnln1)ln(xxexeyxx学习资料总结-名师归纳欢迎下载-欢迎下载 名师归纳-第 5 页，共 5 页 -文档编码:CJ10K3G4Y3M6 HF3R3H1Y1P3 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