implicit differentiation in several variables

Using the multivariable chain rule on $F(x,y)=0$ gives $dy/dx=-F_x/F_y$ directly, without solving for $y$ .

Treating $y$ as a function of $x$ inside $F(x,y)=0$ and differentiating both sides with respect to $x$ gives, by the multivariable chain rule applied to $F(x,y(x))$ , $F_x(x,y) + F_y(x,y)\frac{dy}{dx} = 0 \implies \frac{dy}{dx} = -\frac{F_x(x,y)}{F_y(x,y)}$ wherever $F_y(x,y)\ne 0$ . The formula never requires actually solving $F(x,y)=0$ for $y$ explicitly, which matters since many implicit curves have no clean explicit $y=g(x)$ at all.

Examples

On the folium $x^3+y^3=6xy$ at $(3,3)$ : $F_x=3x^2-6y=9$ , $F_y=3y^2-6x=9$ , so $dy/dx=-9/9=-1$ .