Question

y=1+xe^y求y的二阶导数

Answer 1

两边罩基高锋虚对x求导得：
y'=e^y+xy'e^y
y'=e^y/(1-xe^y)
y''=dy'/dx
=[y'e^y(1-xe^y)-(-e^y-xy'e^y)e^y]/(1-xe^y)
=(2-x)e^(2y)/(1-xe^y)

扩展物尺资料

因为y=1+xe^y，则1-xe^y=2-y，得y'=e^y/(2-y)
即dy/dx=e^y/(2-y)
dy/dx=e^y/(2-y)
==>d(dy/dx)/dx=d(e^y/(2-y))
==>d(dy/dx)/dx=[e^y*dy*(2-y)-e^y*(-dy)]/(2-y)^2
因为dy/dx=e^y/(2-y)，则
==>d(dy/dx)/dx=[e^2y+e^2y/(2-y)]/(2-y)^2
==>d(dy/dx)/dx=e^2y[1+1/(2-y)]/(2-y)^2