设z=z(x,y)由方程xz=lny-lnz所确定的隐函数求偏导z对x和z对y

Question

设z=z(x,y)由方程xz=lny-lnz所确定的隐函数
求偏导 z对x 和 z对y

Answer 1

对x求偏导则把y看成常数
(xz)'=x'*z+x*z'=z+x*z'
(lny-lnz)'=0-(1/z)*z'
z+x*z'=-(1/z)*z'
z'=z/(-x-1/z)=-z²(xz+1)
所以
∂z/∂x=-z²(xz+1)
对y求偏导则把x看成常数
(xz)'=xz'
(lny-lnz)'=1/y-(1/z)*z'
xz'=1/y-(1/z)*z'
z'=(1/y)/(x+1/z)=z/(xyz+y)
所以
∂z/∂y=z/(xyz+y)