1樓:匿名使用者
x+y+z=0 1
xyz=1 2
由12可以確定y=y(x),e69da5e887aa62616964757a686964616f31333433623836z=z(x)
1式兩邊對x求導得1+dy/dx+dz/dx=0 3
2式兩邊對x求導得yz+x(ydz/dx+zdy/dx)=0
即yz+xydz/dx+xzdy/dx=0 4
由34得
dz/dx=(xz-yz)/(xy-xz) 5
dy/dx=(xy-yz)/(xz-xy) 6
d2y/dx2=[(y+xdy/dx-ydz/dx-zdy/dx)(xz-xy)-(xy-yz)(z+xdz/dx-y-xdy/dx)]/(xz-xy)2 7
將56代入7再結合1式,化簡得
d2y/dx2=[y(xz-xy)(x2+y2+z2)-x(xy-yz)(x2+y2+z2)]/(xz-xy)3
=(xyz-xy2-x2y+xyz)(x2+y2+z2)/(xz-xy)3
=[2xyz-xy(y+x)](x2+y2+z2)/(xz-xy)3
=3xyz(x2+y2+z2)/(xz-xy)3
=3(x2+y2+z2)/(xz-xy)3
附(y+xdy/dx-ydz/dx-zdy/dx)(xz-xy)
=y(xz-xy)+x(xy-yz)-y(yz-xz)-z(xy-yz)
=xyz-xy2+x2y-xyz-y2z+xyz-xyz+yz2
=-xy2+x2y-y2z+yz2
=y(x2+z2)-y2(x+z)
=y(x2+z2)+y3
=y(x2+y2+z2)
(xy-yz)(z+xdz/dx-y-xdy/dx)
=(xy-yz)[z+x(xz-yz)/(xy-xz)-y-x(xy-yz)/(xz-xy)]
=(xy-yz)[z(xz-xy)-x(xz-yz)-y(xz-xy)-x(xy-yz)]/(xz-xy)
=(xy-yz)(xz2-xyz-x2z+xyz-xyz+xy2-x2y+xyz)/(xz-xy)
=(xy-yz)(xz2-x2z+xy2-x2y)/(xz-xy)
=(xy-yz)[x(y2+z2)-x2(y+z)]/(xz-xy)
=(xy-yz)[x(y2+z2)+x3]/(xz-xy)
=(xy-yz)[x(x2+y2+z2)]/(xz-xy)
=x(xy-yz)(x2+y2+z2)/(xz-xy)
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