Question

10、已知 z=z(x,y) 由e^z-xyz =0确定，试求 z对x的二次偏导数

10、已知 z=z(x,y) 由e^z-xyz =0确定，试求 z对x的二次偏导数

Answer 1

方程两边对x求偏导数：e^z×αz/αx－yz－xy×αz/αx＝0，所以αz/αx＝yz/(e^z－xy)

二阶偏导数：α^2z/αx^2＝(y×αz/αx×(e^z－xy)－yz(e^z×αz/αx－y))/(e^z－xyz)^2，代入αz/αx＝yz/(e^z－xy)得

α^2z/αx^2＝(2y^2ze^z－2xy^3z－y^2z^2e^z)/(e^z－xy)^3

Answer 2

Fx=-yz,Fy=-xz,Fz=e^z-xy
dz/dx=-Fx/Fz
=yz/(e^z-xy)
d�0�5z/dx�0�5=d(dz/dx)/dx
=-yz*(-y)/(e^z-xy)�0�5
=y�0�5z/(e^z-xy)�0�5