函数y=sin5(19x2+40x+55)的导数计算

函数y=sin5(19x2+40x+55)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=sin5(19x2+40x+55)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=5*sin2(19x2+40x+55)*cos(19x2+40x+55)*(19x2+40x+55)'，=5*sin2(19x2+40x+55)*cos(19x2+40x+55)*(38x+40),=5(38x+40)*sin2(19x2+40x+55)*cos(19x2+40x+55).※.取对数求导计算一阶导数首先对方程两边取对数，有：lny=lnsin5(19x2+40x+55),lny=5lnsin(19x2+40x+55),方程两边同时对x求导，有：y'y=5[sin(19x2+40x+55)]'sin(19x2+40x+55),y'y=5[cos(19x2+40x+55)](38x+40)sin(19x2+40x+55),y'=sin5(19x2+40x+55)*5[cos(19x2+40x+55)](38x+40)sin(19x2+40x+55),y'=sin2(19x2+40x+55)*5[cos(19x2+40x+55)](38x+40),=5(38x+40)sin2(19x2+40x+55)*cos(19x2+40x+55).※.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'=ln5(38x+40)sin2(19x2+40x+55)*cos(19x2+40x+55),则：lny'=ln5+ln(38x+40)+4lnsin(19x2+40x+55)+lncos(19x2+40x+55),对方程两边同时对x再次求导，y''y'=3838x+40+4[sin(19x2+40x+55)]'sin(19x2+40x+55)+[cos(19x2+40x+55)]'cos(19x2+40x+55),=3838x+40+4cos(19x2+40x+55)(38x+40)sin(19x2+40x+55)-sin(19x2+40x+55)(38x+40)cos(19x2+40x+55)=3838x+40+4(38x+40)ctg(19x2+40x+55)-(38x+40)tan(19x2+40x+55),则：y''=5(38x+40)sin2(19x2+40x+55)*cos(19x2+40x+55)[3838x+40+4(38x+40)ctg(19x2+40x+55)-(38x+40)tan(19x2+40x+55)],=190sin2(19x2+40x+55)*cos(19x2+40x+55)+20(38x+40)2sin3(19x2+40x+55)*cos2(19x2+40x+55)-5(38x+40)2sin5(19x2+40x+55),=95sin3(19x2+40x+55)*sin(38x2+80x+110)+20(38x+40)2sin3(19x2+40x+55)*cos2(19x2+40x+55)-5(38x+40)2sin5(19x2+40x+55)。