设y=2arctany/x,求dy/dx,dy^2/d^2x.

设y=2arctan(y/x),求dy/dx,d²y/dx².
设F(x,y)=y-2arctan(y/x)=0,则
dy/dx=-(∂F/∂x)/(∂F/∂y)=-[2(y/x²)/(1+y²/x²)]/[(1-2/x)/(1+y²/x²)]
=-[2y/(x²+y²)]/[x(x-2)/(x²+y²)]=-2y/[x(x-2)]
d²y/dx²=[-2(x²-2x)(dy/dx)+2y(2x-2)]/(x²-2x)²
=[4y+2y(2x-2)]/(x²-2x)²=4(y+xy-y)/(x²-2x)²