y=ln(5x²+2x+9)的导数计算详细步骤

1、※.对数导数计算

∵y=ln(5x²+2x+9),

∴dy/dx=(5x²+2x+9)'/(5x²+2x+9)

=(10x+2)/(5x²+2x+9)。

2、※.导数定义法计算

∵y=ln(5x²+2x+9),

∴dy/dx

=lim(t→0){ln[5(x+t)²+2(x+t)+9]-ln(5x²+2x+9)}/t,

=lim(t→0)ln{[5(x+t)²+2(x+t)+9]/(5x²+2x+9)}/t,

=lim(t→0)ln[(5x²+2x+9+10xt+5t²+2t)/(5x²+2x+9)]/t,

=lim(t→0)ln{1+[(10xt+5t²+2t)/(5x²+2x+9)]^(1/t),

3、=lim(t→0){ln[1+[(10xt+5t²+2t)/(5x²+2x+9)]^[(5x²+2x+9)/(10xt+5t²+2t)]}^[(10xt+5t²+2t)/(5x²+2x+9)t],

=lne^lim(t→0)[(10xt+5t²+2t)/(5x²+2x+9)t],

=lim(t→0)[(10x+5t+2)/(5x²+2x+9)]

=(10x+2)/(5x²+2x+9)。

1、※.函数商的求导

∵dy/dx=(10x+2)/(5x²+2x+9)，

∴d²y/dx²=[10(5x²+2x+9)-(10x+2)(10x+2)]/(5x²+2x+9)²,

=(50x²+20x+90-100x²-40x-4)/(5x²+2x+9)²,

=(-50x²-20x+90-4)/(5x²+2x+9)²,

=-(50x²+20x-86)/(5x²+2x+9)²。

2、※.函数乘积的求导

∵y'=(10x+2)/(5x²+2x+9)

∴(5x²+2x+9)y'=10x+2,两边同时对x求导，有：

(10x+2)y'+(5x²+2x+9)y''=10,

将y'代入上式得：

(10x+2)²/(5x²+2x+9)+(5x²+2x+9)y''=10,

(5x²+2x+9)y''=10-(10x+2)²/(5x²+2x+9),

y''=[10(5x²+2x+9)-(10x+2)²]/(5x²+2x+9)²,

=-(50x²+20x-86)/(5x²+2x+9)²。

1、∵d²y/dx²=-(50x²+20x-86)/(5x²+2x+9)²，

∴d3y/dx3=-[(100x+20)(5x²+2x+9)²-2(50x²+20x-86)(5x²+2x+9)(10x+2)]/(5x²+2x+9)⁴,

=2[(50x²+20x-86)(10x+2)-(50x+10)(5x²+2x+9)]/(5x²+2x+9)²,

=4(125x²+75x²-645x-131)/(5x²+2x+9)².