Z=X2+Y2 X2+2Y2+3Z2=4 求dy/dx,dz/dx

Z=X2+Y2 ①
X2+2Y2+3Z2=4②
①两边对x求导,得
dz/dx=2x+2ydy/dx③
②两边对x求导,得
2x+4ydy/dx+6zdz/dx=0④
③代入④,得
2x+4ydy/dx+6z(2x+2ydy/dx)=0
x+2ydy/dx+6z(x+ydy/dx)=0
(2y+6yz)dy/dx=-x-6xz
dy/dx=(-x-6xz)/(2y+6yz)
从而
dz/dx=2x- (2xy+12xyz)/(2y+6yz)
=(2xy)/(2y+6yz)