如何计算y=ln(5x²+2x+5)的导数

1、※.对数导数计算

∵y=ln(5x²+2x+5),

∴dy/dx=(5x²+2x+5)'/(5x²+2x+5)

=(10x+2)/(5x²+2x+5)。

2、※.导数定义法计算

∵y=ln(5x²+2x+5),

∴dy/dx

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

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

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

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

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

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

=lim(t→0)[(10x+5t+2)/(5x²+2x+5)]

=(10x+2)/(5x²+2x+5)。

1、※.函数商的求导

∵dy/dx=(10x+2)/(5x²+2x+5)，

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

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

=(-50x²-20x+50-4)/(5x²+2x+5)²,

=-(50x²+20x-46)/(5x²+2x+5)²。

2、※.函数乘积的求导

∵y'=(10x+2)/(5x²+2x+5)

∴(5x²+2x+5)y'=10x+2,两边同时对x求导，有：

(10x+2)y'+(5x²+2x+5)y''=10,

3、将y'代入上式得：

(10x+2)²/(5x²+2x+5)+(5x²+2x+5)y''=10,

(5x²+2x+5)y''=10-(10x+2)²/(5x²+2x+5),

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

=-(50x²+20x-46)/(5x²+2x+5)²。

1、∵d²y/dx²=-(50x²+20x-46)/(5x²+2x+5)²，

∴d3y/dx3=-[(100x+20)(5x²+2x+5)²-2(50x²+20x-46)(5x²+2x+5)(10x+2)]/(5x²+2x+5)⁴,

=2[(50x²+20x-46)(10x+2)-(50x+10)(5x²+2x+5)]/(5x²+2x+5)²,

=4(125x²+75x²-345x-71)/(5x²+2x+5)².