题目

隐函数求导y=2x*arctan(y/x)
如题,求dy/dx,及d^2*y/dx^2.要详解,

提问时间：2020-07-24

答案

y=2x*arctan(y/x)
y/x=2*arctan(y/x)
u=y/x
u=2*arctanu
两边求解导数
dy/dx=2arctan(y/x)+2x*1/((y/x)^2+1)*(1/x*dy/dx-y/x^2)
=2arctan(y/x)+2x^3*1/(x^2+y^2)*(1/x*dy/dx-y/x^2)
=2arctan(y/x)+2x^2/(x^2+y^2)*dy/dx-2xy/(x^2+y^2)
(1-2x^2/(x^2+y^2))*dy/dx=2arctan(y/x)-2xy/(x^2+y^2)
(y^2-x^2)/(x^2+y^2)*dy/dx=2arctan(y/x)-2xy/(x^2+y^2)
dy/dx=(x^2+y^2)/(y^2-x^2)*[2arctan(y/x)-2xy/(x^2+y^2)
=2(x^2+y^2)/(y^2-x^2)*arctan(y/x)-2xy/(y^2-x^2)
二阶导数就不计算,太麻烦.
方法是一样的,再两边求解导数