If $F(x,y)=0$, prove $\frac{d^2y}{dx^2}=-\frac{F_{xx}F_y^2-2F_{xy}F_xF_y+F_{yy}F^2_x}{F_y^3}$

Question

If $$F(x,y)=0$$ prove $$\frac{d^2y}{dx^2}=-\frac{F_{xx}F_y^2-2F_{xy}F_xF_y+F_{yy}F^2_x}{F_y^3}$$ I tried $$\frac{dy}{dx}=-\frac{F_x}{F_y}$$ Then $$\frac{d^2y}{dx^2}=-\frac{F_{xx}F_y-F_xF_{xy}}{F_y^2}$$ I do not know where I got wrong... any help? Thanks~

Answer 1

$\frac{d^2y}{dx^2}=-\frac{F_{xx}F_y-F_xF_{xy}}{F_y^2}\quad$ is false.

You got wrong when you confused $\frac{dF_x}{dx}$ with $\frac{\partial F_x}{\partial x}=F_{xx}$ .