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= sin^5 (57x^2+70x+70)一阶和二阶导数的步骤。
方法/步骤
1、※.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：
dy/dx=5*sin^4(57x^2+70x+70)*cos(57x^2+70x+70)*(57x^2+70x+70)’
=5*sin^4 (57x^2+70x+70)*cos(57x^2+70x+70)*(114x+70),
=5(114x+70)*sin^4 (57x^2+70x+70) *cos(57x^2+70x+70).
2、※.取对数求导计算一阶导数
首先对方程两边取对数，有：
lny=lnsin^5 (57x^2+70x+70),
lny=5lnsin(57x^2+70x+70),
3、方程两边同时对x求导，有：
y’/y=5 [sin(57x^2+70x+70)]’/sin(57x^2+70x+70),
y’/y=5 [cos(57x^2+70x+70)](114x+70)/sin(57x^2+70x+70),
y’=sin^5(57x^2+70x+70)*5[cos(57x^2+70x+70)](114x+70)/sin(57x^2+70x+70),
y’=sin^4 (57x^2+70x+70)*5[cos(57x^2+70x+70)](114x+70),
=5 (114x+70)sin^4 (57x^2+70x+70)*cos(57x^2+70x+70).
4、※.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(114x+70)sin^4 (57x^2+70x+70)*cos(57x^2+70x+70),则：
lny’=ln5+ln(114x+70)+4lnsin(57x^2+70x+70)+lncos(57x^2+70x+70),
对方程两边同时对x再次求导，
y’’/y’=114/(114x+70)+4[sin(57x^2+70x+70)]’/sin(57x^2+70x+70)+[cos(57x^2+70x+70)]’/cos(57x^2+70x+70),
=114/(114x+70)+4cos(57x^2+70x+70)(114x+70)/sin(57x^2+70x+70)-sin(57x^2+70x+70)(114x+70)/cos(57x^2+70x+70),
=114/(114x+70)+4(114x+70)ctg(57x^2+70x+70)-(114x+70)tan(57x^2+70x+70),则：
5、y’’=5(114x+70)sin^4(57x^2+70x+70)*cos(57x^2+70x+70)[114/(114x+70)+4(114x+70)ctg(57x^2+70x+70)-(114x+70)tan(57x^2+70x+70)],
=570sin^4(57x^2+70x+70)*cos(57x^2+70x+70)+20(114x+70)^2sin^3(57x^2+70x+70)*cos^2(57x^2+70x+70)-5(114x+70)^2sin^5(57x^2+70x+70),
=285sin^3(57x^2+70x+70)*sin(114x^2+140x+140)+20(114x+70)^2sin^3(57x^2+70x+70)*cos^2(57x^2+70x+70)-5(114x+70)^2sin^5(57x^2+70x+70)。
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