Question

高数求隐函数的偏导数，第二大题的1,2,3,4,题

Answer 1

(1)1+3z²∂z/∂x-y=2∂z/∂x→∂z/∂x=(1-y)/(2-3z²)
2y+3z²∂z/∂y-x=2∂z/∂y→∂z/∂y=(2y-x)/(2-3z²)
(2)e^x+e^z∂z/∂x=3yz+3xy∂z/∂x→∂z/∂x=(e^x-3yz)/(3xy-e^z)
e^y+e^z∂z/∂y=3xz+3xy∂z/∂y→∂z/∂y=(e^y-3xz)/(3xy-e^z)
(3)∂z/∂x=1/(x+y)-e^z∂z/∂x→∂z/∂x=1/[(x+y)·(1+e^z)]
∂z/∂y=1/(x+y)-e^z∂z/∂y→∂z/∂y=1/[(x+y)·(1+e^z)]
z=1-e^z→z=0
Zx(1,0)=1/2=Zy(1,0)
(4)e^z∂z/∂x-yz-xy∂z/∂x=0
∂z/∂x=yz/(e^z-xy)
∂z/∂y=xz/(e^z-xy)
∂²z/∂x²=[y∂z/∂x·(e^z-xy)-yz·(e^z∂z/∂x-y)]/(e^z-xy)²
=[2y²z(e^z-xy)-y²z²e^z]/(e^z-xy)³
∂²z/∂x∂y=[(z+yxz/(e^z-xy))·(e^z-xy)-yz·(e^z·xz/(e^z-xy)-x)]/(e^z-xy)²
=[z(e^z-xy)²+2xyz(e^z-xy)-xyz²]/(e^z-xy)³