设z=z(x,y)是由方程x*x+y*y+z*z-2z=0,求z对x的二阶偏导

Question

设z=z(x,y)是由方程x*x+y*y+z*z-2z=0,求z对x的二阶偏导

Answer 1

x²+y²+z²-2z=0
两边对x求偏导：2x+2zZ'x-2Z'x=0,得：Z'x=x/(1-z)
再对Z'x求偏导：Z"xx=[1-z+xZ'x]/(1-z)²=[1-z+x²/(1-z)]/(1-z)²=[(1-z)²+x²]/(1-z)³