函数y=ln(5x²+2x+7)的多阶导数计算

1、※.对数导数计算

∵y=ln(5x²+2x+7),

∴dy/dx=(5x²+2x+7)'/(5x²+2x+7)

=(10x+2)/(5x²+2x+7)。

2、※.导数定义法计算

∵y=ln(5x²+2x+7),

∴dy/dx

=lim(t→0){ln[5(x+t)²+2(x+t)+7]-ln(5x²+2x+7)}/t,

=lim(t→0)ln{[5(x+t)²+2(x+t)+7]/(5x²+2x+7)}/t,

=lim(t→0)ln[(5x²+2x+7+10xt+5t²+2t)/(5x²+2x+7)]/t,

=lim(t→0)ln{1+[(10xt+5t²+2t)/(5x²+2x+7)]^(1/t),

3、=lim(t→0){ln[1+[(10xt+5t²+2t)/(5x²+2x+7)]^[(5x²+2x+7)/(10xt+5t²+2t)]}^[(10xt+5t²+2t)/(5x²+2x+7)t],

=lne^lim(t→0)[(10xt+5t²+2t)/(5x²+2x+7)t],

=lim(t→0)[(10x+5t+2)/(5x²+2x+7)]

=(10x+2)/(5x²+2x+7)。

1、※.函数商的求导

∵dy/dx=(10x+2)/(5x²+2x+7)，

∴d²y/dx²=[10(5x²+2x+7)-(10x+2)(10x+2)]/(5x²+2x+7)²,

=(50x²+20x+70-100x²-40x-4)/(5x²+2x+7)²,

=(-50x²-20x+70-4)/(5x²+2x+7)²,

=-(50x²+20x-66)/(5x²+2x+7)²。

2、※.函数乘积的求导

∵y'=(10x+2)/(5x²+2x+7)

∴(5x²+2x+7)y'=10x+2,两边同时对x求导，有：

(10x+2)y'+(5x²+2x+7)y''=10,

将y'代入上式得：

(10x+2)²/(5x²+2x+7)+(5x²+2x+7)y''=10,

(5x²+2x+7)y''=10-(10x+2)²/(5x²+2x+7),

y''=[10(5x²+2x+7)-(10x+2)²]/(5x²+2x+7)²,

=-(50x²+20x-66)/(5x²+2x+7)²。

1、∵d²y/dx²=-(50x²+20x-66)/(5x²+2x+7)²，

∴d3y/dx3=-[(100x+20)(5x²+2x+7)²-2(50x²+20x-66)(5x²+2x+7)(10x+2)]/(5x²+2x+7)⁴,

=2[(50x²+20x-66)(10x+2)-(50x+10)(5x²+2x+7)]/(5x²+2x+7)²,

=4(125x²+75x²-495x-101)/(5x²+2x+7)².