z∧3-3xyz＝a∧3,求先对x偏导再对y的二阶偏导.答案是z（z∧4-2xyz∧2-x∧2y∧2）/（z∧2...z∧3-3xyz＝a∧3,求先对x偏导再对y的二阶偏导.答案是z（z∧4-2xyz∧2-x∧2y∧2）/（z∧2-xy）∧3..我解了很多遍

Question

z∧3-3xyz＝a∧3,求先对x偏导再对y的二阶偏导.答案是z（z∧4-2xyz∧2-x∧2y∧2）/（z∧2...
z∧3-3xyz＝a∧3,求先对x偏导再对y的二阶偏导.答案是z（z∧4-2xyz∧2-x∧2y∧2）/（z∧2-xy）∧3..我解了很多遍就是没有解到z∧4?

Answer 1

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