Question

Consider F(x,y)=0, which implicitly describes a curve in the plane. We use the modifier "implicitly" since we may not know how y is explicitly described as a function of x. We may, however, assume that y=y(x), though we may never know this function explicitly. So we can consider our curve as described by F(x,y(x))=0. By differentiating both sides of this expression using the First Version of the Chain Rule, show that dy/dx=-(∂F/∂x)/(∂F/∂y)

Answer 1

just take the derivative and call it a day:\[F(x,y)=0 ............\delta F(x,y) =\frac{\delta F} {\delta x} dx + \frac{\delta F }{\delta y} dy = 0\]

\[\frac{ \delta F}{\delta y} dy=-\frac{\delta F}{\delta x} dx\]\[\frac{dy}{dx}=- \frac{\frac{\delta F}{\delta x}} {\frac {\delta F}{\delta y}}\]

Answer 2

i hope this helps