设e^z=xyz,求d^2z/dxdy用多元函数方法求隐函数的二阶导数求详解F=e^z-xyzFx=-yzFy=-xzFz=e^z-xy.

Question

设e^z=xyz,求d^2z/dxdy用多元函数方法求隐函数的二阶导数 求详解
F=e^z-xyz
Fx=-yz Fy=-xz Fz=e^z-xy
.

Answer 1

F(x,y,z)=e^z-xyz=0
dz/dx=-Fx/Fz,dz/dy=-Fy/Fz
Fx=-yz; Fy=-xz; Fz=e^z-xy
dz/dx=yz/(e^z-xy);
dz/dy=xz/(e^z-xy);
d^2z/dxdy=d(dz/dx)/dy
=d(-Fx/Fz)/dy+d(-Fx/Fz)/dz*dz/dy
=[z(e^z-xy)-yz*(-x)]/(e^z-xy)^2+[y(e^z-xy)-yz*e^z]/(e^z-xy)^2*xz/(e^z-xy)
=ze^z/(e^z-xy)^2+xyz(e^z-ze^z-xy)/(e^z-xy)^3
=[ze^(2z)-xyze^z+xyz(e^z-ze^z-xy)]/(e^z-xy)^3
=[ze^(2z)-xyz(ze^z+xy)]/(e^z-xy)^3