y=ln(5x²+2x+2)的导数计算

1、一阶导数：

※.对数导数计算

∵y=ln(5x²+2x+2),

∴dy/dx=(5x²+2x+2)'/(5x²+2x+2)

=(10x+2)/(5x²+2x+2)。

2、※.导数定义法计算

∵y=ln(5x²+2x+2),

∴dy/dx

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

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

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

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

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

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

=lim(t→0)[(10x+5t+2)/(5x²+2x+2)]

=(10x+2)/(5x²+2x+2)。

3、二阶导数计算

※.函数商的求导

∵dy/dx=(10x+2)/(5x²+2x+2)，

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

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

=(-50x²-20x+20-4)/(5x²+2x+2)²,

=-(50x²+20x-16)/(5x²+2x+2)²。

4、※.函数乘积的求导

∵y'=(10x+2)/(5x²+2x+2)

∴(5x²+2x+2)y'=10x+2,两边同时对x求导，有：

(10x+2)y'+(5x²+2x+2)y''=10,

将y'代入上式得：

(10x+2)²/(5x²+2x+2)+(5x²+2x+2)y''=10,

(5x²+2x+2)y''=10-(10x+2)²/(5x²+2x+2),

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

=-(50x²+20x-16)/(5x²+2x+2)²。

5、三阶导数计算：

∵d²y/dx²=-(50x²+20x-16)/(5x²+2x+2)²，

∴d³y/dx²=-[(100x+20)(5x²+2x+2)²-2(50x²+20x-16)(5x²+2x+2)(10x+2)]/(5x²+2x+2)⁴,

=2[(50x²+20x-16)(10x+2)-(50x+10)(5x²+2x+2)]/(5x²+2x+2)²,

=4(125x²+75x²-120x-26)/(5x²+2x+2)².