设y=f(x,t),而t=t(x,y)可由F(x,y,t)=0确定,求dy/dx.

题目

设y=f(x,t),而t=t(x,y)可由F(x,y,t)=0确定,求dy/dx.
f,F都是可微函数

答案

∂
dy/dx= ∂f/ ∂x + (∂f/ ∂t)(∂t/∂x)
dF/dx= (∂F/∂t)(∂t/ ∂x)+ (∂F/∂y)(∂y/ ∂x) + (∂F/∂x)=0
dy/dx= ∂f/∂x +(∂f/∂t)[dF/dx-(∂F/∂y)(∂y/∂x)-(∂F/∂x)]/(∂F/∂t)