Question

设e^z-xyz=0,求z对x的二阶偏导

Answer 1

简单计算一下即可，答案如图所示

Answer 2

e^z-xyz=0 方程两边对x求偏导
e^z(∂z/∂x)-yz-xy(∂z/∂x)=0
得∂z/∂x=yz/(e^z-xy) .........(1)
对(1)按运算法则继续对x求导
∂^2z/∂x^2=[y(∂z/∂x)(e^z-xy)-yz(e^z(∂z/∂x)-y)]/(e^z-xy)^2 .......(2)
将(1)代入(2)化简得：
=[2zy^2(e^z-xy)-(e^z)*(y^2)*(z^2)]/(e^z-xy)^3

Answer 3

设方程e的z次方-xyz=0确定函数z=(fx,y) 求z对x的二阶偏导数

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)]

Answer 4

第一个答案是对的，跟课后答案一样