高数复杂隐函数的二阶导数

Question

如题所述

Answer 1

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Answer 2

两边取对数：
ln(x^2+y^2)=2arctan(y/x) 两边对x求导：
(2x+2yy')/(x^2+y^2)=2(xy'-y)/(1+(y^2/x^2))]
x+yy'=x^2(xy'-y)=x^3y'-x^2y
x+yy'=x^3y'-x^2y
y'(y-x^3)=-x^2y-x
两边对x求导：

y''(y-x^3)+y'(y'-3x^2)=-x^2y'-2xy-1

y''=[2x^2y'-(y')^2-2xy-1]/(y-x^3) 其中：y'=-(x^2y+x)/(y-x^3)