函数y=sin^7(49x^2+51x+64)的导数计算

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^7 (49x^2+51x+64)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=7*sin^6(49x^2+51x+64)*cos(49x^2+51x+64)*(49x^2+51x+64)’

=7*sin^6 (49x^2+51x+64)*cos(49x^2+51x+64)*(98x+51),

=7(98x+51)*sin^6 (49x^2+51x+64) *cos(49x^2+51x+64).

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^7 (49x^2+51x+64),

lny=7lnsin(49x^2+51x+64),

3、方程两边同时对x求导，有：

y’/y=7 [sin(49x^2+51x+64)]’/sin(49x^2+51x+64),

y’/y=7 [cos(49x^2+51x+64)](98x+51)/sin(49x^2+51x+64),

y’=sin^7(49x^2+51x+64)*7[cos(49x^2+51x+64)](98x+51)/sin(49x^2+51x+64),

y’=sin^6 (49x^2+51x+64)*7[cos(49x^2+51x+64)](98x+51),

=7 (98x+51)sin^6 (49x^2+51x+64)*cos(49x^2+51x+64).

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’=ln7(98x+51)sin^6 (49x^2+51x+64)*cos(49x^2+51x+64),则：

lny’=ln7+ln(98x+51)+6lnsin(49x^2+51x+64)+lncos(49x^2+51x+64),

5、对方程两边同时对x再次求导，

y’’/y’=98/(98x+51)+6[sin(49x^2+51x+64)]’/sin(49x^2+51x+64)+[cos(49x^2+51x+64)]’/cos(49x^2+51x+64),

=98/(98x+51)+6cos(49x^2+51x+64)(98x+51)/sin(49x^2+51x+64)-sin(49x^2+51x+64)(98x+51)/cos(49x^2+51x+64),

=98/(98x+51)+6(98x+51)ctg(49x^2+51x+64)-(98x+51)tan(49x^2+51x+64),则：

6、y’’=7(98x+51)sin^6(49x^2+51x+64)*cos(49x^2+51x+64)[98/(98x+51)+6(98x+51)ctg(49x^2+51x+64)-(98x+51)tan(49x^2+51x+64)],

=686sin^6(49x^2+51x+64)*cos(49x^2+51x+64)+42(98x+51)^2sin^5(49x^2+51x+64)*cos^2(49x^2+51x+64)-7(98x+51)^2sin^7(49x^2+51x+64),

=343sin^5(49x^2+51x+64)*sin(98x^2+102x+128)+42(98x+51)^2sin^5(49x^2+51x+64)*cos^2(49x^2+51x+64)-7(98x+51)^2sin^7(49x^2+51x+64)。

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