函数y=sin^6(44x^2+28x+36)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^6 (44x^2+28x+36)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=6*sin^5(44x^2+28x+36)*cos(44x^2+28x+36)*(44x^2+28x+36)’

=6*sin^5 (44x^2+28x+36)*cos(44x^2+28x+36)*(88x+28),

=6(88x+28)*sin^5 (44x^2+28x+36) *cos(44x^2+28x+36).

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^6 (44x^2+28x+36),

lny=6lnsin(44x^2+28x+36),

3、方程两边同时对x求导,有:

y’/y=6 [sin(44x^2+28x+36)]’/sin(44x^2+28x+36),

y’/y=6 [cos(44x^2+28x+36)](88x+28)/sin(44x^2+28x+36),

y’=sin^6(44x^2+28x+36)*6[cos(44x^2+28x+36)](88x+28)/sin(44x^2+28x+36),

y’=sin^5 (44x^2+28x+36)*6[cos(44x^2+28x+36)](88x+28),

=6 (88x+28)sin^5 (44x^2+28x+36)*cos(44x^2+28x+36).

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’=ln6(88x+28)sin^5 (44x^2+28x+36)*cos(44x^2+28x+36),则:

lny’=ln6+ln(88x+28)+5lnsin(44x^2+28x+36)+lncos(44x^2+28x+36),

5、对方程两边同时对x再次求导,

y’’/y’=88/(88x+28)+5[sin(44x^2+28x+36)]’/sin(44x^2+28x+36)+[cos(44x^2+28x+36)]’/cos(44x^2+28x+36),

=88/(88x+28)+5cos(44x^2+28x+36)(88x+28)/sin(44x^2+28x+36)-sin(44x^2+28x+36)(88x+28)/cos(44x^2+28x+36),

=88/(88x+28)+5(88x+28)ctg(44x^2+28x+36)-(88x+28)tan(44x^2+28x+36),则:

6、y’’=6(88x+28)sin^5(44x^2+28x+36)*cos(44x^2+28x+36)[88/(88x+28)+5(88x+28)ctg(44x^2+28x+36)-(88x+28)tan(44x^2+28x+36)],

=528sin^5(44x^2+28x+36)*cos(44x^2+28x+36)+30(88x+28)^2sin^4(44x^2+28x+36)*cos^2(44x^2+28x+36)-6(88x+28)^2sin^6(44x^2+28x+36),

=264sin^4(44x^2+28x+36)*sin(88x^2+56x+72)+30(88x+28)^2sin^4(44x^2+28x+36)*cos^2(44x^2+28x+36)-6(88x+28)^2sin^6(44x^2+28x+36)。

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