Concept preview

What you will be able to do

1. 01Use implicit differentiation to find $\dfrac{dy}{dx}$ for a relation, treating $y$ as a function of $x$ so every $y$-term carries a $\dfrac{dy}{dx}$ factor.
2. 02Locate HORIZONTAL tangents where the numerator of $\dfrac{dy}{dx}$ is zero and VERTICAL tangents where the denominator is zero, checking that the point lies on the curve.
3. 03Compute the second derivative $\dfrac{d^2y}{dx^2}$ of an implicit relation and use its sign to determine concavity of curves such as $x^2+y^2=r^2$, $xy=k$, and ellipses.

The governing idea of implicit differentiation is a single sentence: along the curve, $y$ is a function of $x$, so differentiating any term that contains $y$ requires the chain rule and produces a $\dfrac{dy}{dx}$ factor. Differentiating $x^2$ gives $2x$; differentiating $y^2$ gives $2y\dfrac{dy}{dx}$ — NOT just $2y$. The $\dfrac{dy}{dx}$ is the price of the chain rule, and it is exactly what a careless solver drops.