求有方程y=x+ln y所确定的函数y=y(x)的微分dy

题目

求有方程y=x+ln y所确定的函数y=y(x)的微分dy

答案

F(x,y)=x+lny-y=0
dF(x,y)=0=(∂F(x,y)dx/∂x)+(∂F(x,y)dy/∂y)
dy/dx=-(∂F(x,y)/∂x)/(∂F(x,y)/∂y)
=-1/(1/y-1)
=y/(y-1)
dy=ydx/(y-1)