设方程e的z次方-xyz=0确定函数z=(fx,y)求对x的二阶偏导数

题目

设方程e的z次方-xyz=0确定函数z=(fx,y)求对x的二阶偏导数

答案

e^z - xyz = 0
e^z(∂z/∂x) = yz + xy(∂z/∂x)
令z' = ∂z/∂x = yz/(e^z - xy) = yz/(xyz - xy) = z/(xz-x) = [z/(z-1)](1/x)
∂²z/∂x²
= dz'/dx
= (1/x)[z'(z-1)-zz']/(z-1)² - (1/x²)[z/(z-1)]
= -z'/[x(z-1)²] - z/[(z-1)x²]
将z'代入就有
∂²z/∂x² = -z/[x²(z-1)³] - z/[(z-1)x²] = -(z/x²)[1/(z-1)³ + 1/(z-1)]