Question

设方程F(x,y)=0确定隐函数y=f(x),且F(x,y)存在二阶连续偏导数,求其二阶导数

答案应该是一串没有具体数字全是符号的式子

Answer 1

F(x,y)存在二阶连续偏导数且对 y 的偏导数不为 0，求 y 的二阶导数？
将等式 F(x,y)=0 两边对 x 求导：∂F/∂x +(∂F/∂y)(dy/dx)=0，∴ y'=dy/dx=-(∂F/∂x)/(∂F/∂y)；
y"=dy'/dx=d[-(∂F/∂x)/(∂F/∂y)]/dx
=-[(∂²F/∂x²)*(∂F/∂y)-(∂F/∂x)*(∂²F/∂x² +∂²F/∂x∂y*y')]/(∂F/∂y)²
=-[(∂²F/∂x²)(∂F/∂y)-(∂F/∂x)(∂²F/∂x²) +(∂F/∂x)²*(∂²F/∂x∂y)/(∂F/∂y)]/(∂F/∂y)²
=-[(∂²F/∂x²)-(∂F/∂y)(∂F/∂x)(∂²F/∂x²) +(∂F/∂x)²*(∂²F/∂x∂y)]/(∂F/∂y)³；