Chapter 12 - Functions of Several Veriables - 12.5 The Chain Rule - 12.5 Exercises - Page 913: 6

$$\frac{dy}{dx} =- \frac{\partial F}{\partial x}/\frac{\partial F}{\partial y}.$$

By differentiation with respect to $x$, we have $$ \frac{\partial F}{\partial x} + \frac{\partial F}{\partial y} \frac{dy}{dx} =0. $$That is $$\frac{dy}{dx} =- \frac{\partial F}{\partial x}/\frac{\partial F}{\partial y}.$$