Section 14.5 Chain Rule

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 \(F_y \ne 0\). Prove that if \(F\) has continuous second derivatives, then a formula for the second derivative of \(y\) is \[ \frac{d^2 y}{dx^2} = - \frac{F_{xx}F_y^2 - 2F_{xy}F_x F_y + F_{yy}F_x^2}{F_y^3} \]