Question

高数，隐函数求导法，设z³-3xyz＝a³，求σ²z/σxσy求详细过程，感激

Answer 1

对x求导：3z^2 z'-3yz-3xyz'=0
得：z'=yz/(z^2-xy)
再对x求导：
z"=y[z'(z^2-xy)-z(2zz')]/(z^2-xy)^2
=-yz'(z^2+xy)/(z^2-xy)^2
=-y^2 z/(z^2-xy)^3

运用隐函数求导法则,两端对x求导得
用隐函数微分法
令F[x,y,z] = z³-3xyz-a³
z'x = -F'x/F'z = yz/(z²-xy)
z'y = -F'y/F'z = xz/(z²-xy)
z''xy = [z'x]'y = [(yz)'(z² - xy) - yz * (2z z'y - x)]/(z²-xy)²
= [(z + y z'y)(z²-xy) - 2yz² z'y + xyz]/(z²-xy)²
= (z³ - yz² z'y - xy² z'y)/(z²-xy)²
= [z³ - (yz²+xy²)xz/(z²-xy)]/(z²-xy)²
= z(z^4 - 2xyz³ - x²y²z)/(z²-xy)³

Answer 2

简单计算一下即可，答案如图所示