{高数} 隐函数求导题两道
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{高数} 隐函数求导题两道
1、设z^3-3xyz=a^3,求:ð^2z/(ðxðy).
2、设e^z-xyz=0,求:ð^2z/ð^x.
根据隐函数求导定理中给出的公式做到一阶导数(有分子分母的函数)后就不晓得怎么求二阶导数了.
1、设z^3-3xyz=a^3,求:ð^2z/(ðxðy).
2、设e^z-xyz=0,求:ð^2z/ð^x.
根据隐函数求导定理中给出的公式做到一阶导数(有分子分母的函数)后就不晓得怎么求二阶导数了.
![{高数} 隐函数求导题两道](/uploads/image/z/154255-31-5.jpg?t=%7B%E9%AB%98%E6%95%B0%7D+%E9%9A%90%E5%87%BD%E6%95%B0%E6%B1%82%E5%AF%BC%E9%A2%98%E4%B8%A4%E9%81%93)
1.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也是y的函数,刚才我当成常数扔了- -!)
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
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也是y的函数,刚才我当成常数扔了- -!)
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