= Implicit differentiation
{wiki}
Implicit differentiation differentiates an identity $F(x,y(x))=0$ along a <differentiable> branch and uses the <chain rule> to solve for its <derivative>:
$$
F_x+F_y\,y'=0,\qquad y'=-\frac{F_x}{F_y}\quad(F_y\ne0).
$$
The <partial derivatives> $F_x,F_y$ treat $x,y$ as independent coordinates before evaluating on the branch. The <implicit function theorem> justifies the branch's local existence and regularity under its hypotheses; differentiating alone does not establish either. For the <unit circle> at $(3/5,4/5)$, this calculation yields the <tangent line> $y-4/5=-(3/4)(x-3/5)$.
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