Question

设函数z=z(x,y),由方程x+ln(xy-z)=y^-1+z所确定,求全微分dz(x,y,z)=(1,1,0)

Answer 1

两边先对x求偏导
dx+(xy-z)'*1/(xy-z)=dz
dx+(ydx-dz)/(xy-z)=dz
dx(xy-z)+ydx-dz=dz(xy-z)
xydx-zdx+ydx-dz=dz(xy-z)
dx(xy-z+y)=dz(xy-z+1)
dz/dx=(xy-z+y)/(xy-z+1)
再对y求偏导
(xy-z)'*1/(xy-z)=-y^(-2)dy+dz
(xdy-dz)/(xy-z)=-dy/y²+dz
xdy-dz=-(xy-z)dy/y²+dz(xy-z)
dy(x+(xy-z)/y²)dy=dz(xy-z+1)
dz/dy=(x+(xy-z)/y²)/(xy-z+1)

dz=(xy-z+y)dx/(xy-z+1)+(x+(xy-z)/y²)dy/(xy-z+1)
把(1,1,0)代入
dz=dx+dy