Differentiation: Implicit Functions

Suppose we know that $$x^2+y^2=1,$$ and want to calculate gradients on this curve. Note, using the Chain Rule, that $$\frac{d}{dx}(y^2)=\frac{d}{dy}(y^2)\times\frac{dy}{dx}.$$ It follows that if we differentiate our curve equation with respect to $x$ we obtain $$2x+2y\frac{dy}{dx}=0,$$ from which we get that $$\frac{dy}{dx}=-x/y.$$ This technique is known as implicit differentiation. Using it, it's easy to prove results such as $$\frac{dx}{dy}=\frac{1}{dy/dx}.$$