函数y=sin4(55x2+11x+34)的导数计算

函数y=sin4(55x2+11x+34)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=sin4(55x2+11x+34)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=4*sin2(55x2+11x+34)*cos(55x2+11x+34)*(55x2+11x+34)'，=4*sin2(55x2+11x+34)*cos(55x2+11x+34)*(110x+11),=4(110x+11)*sin2(55x2+11x+34)*cos(55x2+11x+34).※.取对数求导计算一阶导数首先对方程两边取对数，有：lny=lnsin4(55x2+11x+34),lny=4lnsin(55x2+11x+34),方程两边同时对x求导，有：y'y=4[sin(55x2+11x+34)]'sin(55x2+11x+34),y'y=4[cos(55x2+11x+34)](110x+11)sin(55x2+11x+34),y'=sin4(55x2+11x+34)*4[cos(55x2+11x+34)](110x+11)sin(55x2+11x+34),y'=sin2(55x2+11x+34)*4[cos(55x2+11x+34)](110x+11),=4(110x+11)sin2(55x2+11x+34)*cos(55x2+11x+34).※.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'=ln4(110x+11)sin2(55x2+11x+34)*cos(55x2+11x+34),则：lny'=ln4+ln(110x+11)+3lnsin(55x2+11x+34)+lncos(55x2+11x+34),对方程两边同时对x再次求导，y''y'=110110x+11+3[sin(55x2+11x+34)]'sin(55x2+11x+34)+[cos(55x2+11x+34)]'cos(55x2+11x+34),=110110x+11+3cos(55x2+11x+34)(110x+11)sin(55x2+11x+34)-sin(55x2+11x+34)(110x+11)cos(55x2+11x+34)=110110x+11+3(110x+11)ctg(55x2+11x+34)-(110x+11)tan(55x2+11x+34),则：y''=4(110x+11)sin2(55x2+11x+34)*cos(55x2+11x+34)[110110x+11+3(110x+11)ctg(55x2+11x+34)-(110x+11)tan(55x2+11x+34)],=440sin2(55x2+11x+34)*cos(55x2+11x+34)+12(110x+11)2sin2(55x2+11x+34)*cos2(55x2+11x+34)-4(110x+11)2sin4(55x2+11x+34),=220sin2(55x2+11x+34)*sin(110x2+22x+68)+12(110x+11)2sin2(55x2+11x+34)*cos2(55x2+11x+34)-4(110x+11)2sin4(55x2+11x+34)。