Textbook Problem

Equation 6 is a formula for the derivative dy/dx of a function defined implicitly by an equation F(x, y) = 0, provided that F is differentiable and F_y ≠ 0. Prove that if F has continuous second derivatives, then a formula for the second derivative of y is

$\frac{d^{2}y}{d{x}^2}=-\frac{F_{xx}{F}^2{}_y-2F_{xy}{F}_x{F}_y+F_{yy}{F}^2{}_x}{F^3{}_y}$