Implicit Differentiation

The central idea

An equation $F(x,y)=0$ may define $y$ locally as a differentiable function of $x$ when $F$ is differentiable and $F_y\ne0$. Differentiating along the curve gives $F_x+F_y y'=0$, hence $y'=-F_x/F_y$. Every derivative of a $y$-expression must include $dy/dx$ by the chain rule.

Definitions, hypotheses, and notation

The formula $y'=-F_x/F_y$ exposes a boundary case: where $F_y=0$, solving locally for $y$ as a differentiable function of $x$ may fail. A curve may instead have a vertical tangent, and it may be possible to solve for $x$ as a function of $y$ if $F_x\ne0$. The algebraic denominator therefore has geometric meaning.

Second derivatives require another round of implicit differentiation and product rules because $y'$ also depends on $x$. It is usually clearest to isolate $y'$ first, differentiate that relation, and only then substitute a point. Even for first derivatives, substitution after differentiation preserves which quantities vary along the curve.

Conceptual meaning

Implicit differentiation finds tangent slopes without first solving for $y$. The gradient $\nabla F=\langle F_x,F_y\rangle$ is normal to the level curve, while the tangent direction $\langle1,y'\rangle$ is perpendicular to it.

A dependable method and decision rule

1. Differentiate both sides with respect to $x$.
2. Treat $y$ as the composite function $y(x)$.
3. Attach a factor $y'$ whenever differentiating a function of $y$.
4. Collect all terms containing $y'$ on one side and factor it out.
5. Solve for $y'$ before substituting the coordinates of a requested point.

Fully worked example

Common mistakes and why they fail