Question

多元函数微分的问题？

怎么用隐函数求导法则dy/dx=-DX/DY来做？

Answer 1

z=x^2+y^2
dz/dx = 2x + 2y.dy/dx (1)
//
x^2+2y^2+3z^2 =20
2x+2y.dy/dx +6z.dz/dx =0 (2)
from (1) and (2)
2x+2y.dy/dx +6z.( 2x + 2y. dy/dx) =0
( 2x + 2y. dy/dx) . (1+6z) =0
2x + 2y. dy/dx=0
dy/dx= -x/y
from (1)
dz/dx
= 2x + 2y.dy/dx
=2x +2y( -x/y)
=2x -2x
=0

Answer 2

z=x^2+y^2
dz/dx = 2x + 2y.dy/dx (1)
x^2+2y^2+3z^2 =20
2x+4y.dy/dx +6z.dz/dx =0
x+2ydy/dx+3zdz/dx=0.(2)
把(1)代入(2),得x+2ydy/dx+3z(2x+2ydy/dx)=0,
整理得(2y+6yz)dy/dx=-x-6xz,
所以dy/dx=(-x-6xz)/(2y+6yz),
代入(1),得dz/dx=2x+(-x-6xz)/(1+3z)=x/(1+3z).