函数y=sin9(31x2+40x+71)的导数计算

函数y=sin9(31x2+40x+71)的导数计算主要内容：本文主要用复合函数求导法则、链式求导法则以及取对数求导等方法，介绍计算函数y=sin9(31x2+40x+71)一阶和二阶导数的步骤。※.复合函数链式求导计算一阶导数由复合函数求导法则，对x求导有：dydx=9*sin2(31x2+40x+71)*cos(31x2+40x+71)*(31x2+40x+71)'，=9*sin2(31x2+40x+71)*cos(31x2+40x+71)*(62x+40),=9(62x+40)*sin2(31x2+40x+71)*cos(31x2+40x+71).※.取对数求导计算一阶导数首先对方程两边取对数，有：lny=lnsin9(31x2+40x+71),lny=9lnsin(31x2+40x+71),方程两边同时对x求导，有：y'y=9[sin(31x2+40x+71)]'sin(31x2+40x+71),y'y=9[cos(31x2+40x+71)](62x+40)sin(31x2+40x+71),y'=sin9(31x2+40x+71)*9[cos(31x2+40x+71)](62x+40)sin(31x2+40x+71),y'=sin2(31x2+40x+71)*9[cos(31x2+40x+71)](62x+40),=9(62x+40)sin2(31x2+40x+71)*cos(31x2+40x+71).※.二阶导数计算本处根据函数特征，采取取对数计算导数，首先对函数两边同时取对数，有：lny'=ln9(62x+40)sin2(31x2+40x+71)*cos(31x2+40x+71),则：lny'=ln9+ln(62x+40)+8lnsin(31x2+40x+71)+lncos(31x2+40x+71),对方程两边同时对x再次求导，y''y'=6262x+40+8[sin(31x2+40x+71)]'sin(31x2+40x+71)+[cos(31x2+40x+71)]'cos(31x2+40x+71),=6262x+40+8cos(31x2+40x+71)(62x+40)sin(31x2+40x+71)-sin(31x2+40x+71)(62x+40)cos(31x2+40x+71)=6262x+40+8(62x+40)ctg(31x2+40x+71)-(62x+40)tan(31x2+40x+71),则：y''=9(62x+40)sin2(31x2+40x+71)*cos(31x2+40x+71)[6262x+40+8(62x+40)ctg(31x2+40x+71)-(62x+40)tan(31x2+40x+71)],=558sin2(31x2+40x+71)*cos(31x2+40x+71)+72(62x+40)2sin7(31x2+40x+71)*cos2(31x2+40x+71)-9(62x+40)2sin9(31x2+40x+71),=279sin7(31x2+40x+71)*sin(62x2+80x+142)+72(62x+40)2sin7(31x2+40x+71)*cos2(31x2+40x+71)-9(62x+40)2sin9(31x2+40x+71)。