5.12 Exploring Behaviors of Implicit Relations

Implicit Differentiation for Slopes

For curves defined implicitly (not solved for y), differentiate both sides with respect to x, treating y as a function of x and applying the chain rule to every y-term. Solve for dy/dx to get the slope. This lets you analyze tangent lines, increasing/decreasing behavior, and extrema even when you can't isolate y.

Second Derivatives Implicitly

To find concavity for implicit relations, differentiate dy/dx again with respect to x, substituting the expression for dy/dx you already found. Simplify using the original equation when possible. Set d²y/dx² = 0 (and check sign changes) to locate points of inflection.

Finding Extrema and Tangent Lines

Horizontal tangents occur where dy/dx = 0 (numerator zero, denominator nonzero); vertical tangents occur where dy/dx is undefined (denominator zero, numerator nonzero). Use the original implicit equation to find corresponding y-values, and classify extrema by checking sign changes in dy/dx or using the second derivative test.

Key terms

Implicit differentiation

Differentiating both sides of an equation with respect to x, treating y as a function of x and applying the chain rule to y-terms.

Horizontal tangent condition

dy/dx = 0, with numerator zero and denominator nonzero.

Vertical tangent condition

dy/dx is undefined, with denominator zero and numerator nonzero.

Finding d²y/dx² implicitly

Differentiate dy/dx again with respect to x, substitute the known expression for dy/dx, and simplify using the original equation.

Point of inflection (implicit curve)

A point where the second derivative equals zero and concavity changes sign.

Try an AP-style question

Where does x² + y² = 16 have horizontal tangent lines?

1. (0, 4) and (0, −4)
2. Nowhere
3. (0, 0) only
4. (4, 0) and (−4, 0)