Implicit Relations and Behavior
An implicit curve is defined by an equation of the form F(x, y) = 0, where neither variable is explicitly solved for in terms of the other. Classic examples include the unit circle x² + y² = 1, the folium of Descartes x³ + y³ = 3xy, and the lemniscate of Bernoulli (x² + y²)² = x² − y². Unlike explicit functions y = f(x), implicit curves can loop back on themselves, cross, or form closed shapes — behaviors that no single explicit function can capture. Understanding how these curves behave requires the technique of implicit differentiation. By differentiating both sides of F(x, y) = 0 with respect to x and treating y as a function of x via the chain rule, we obtain dy/dx without ever solving for y explicitly. This gives us access to slopes of tangent lines, locations of horizontal and vertical tangents, and information about concavity — all crucial for sketching and analyzing the curve's behavior. Applications of implicit differentiation appear throughout physics, engineering, and economics. Contour lines on a topographic map, level curves of a potential function, and constraint curves in optimization problems are all implicit relations. Mastering the analysis of implicit curves — finding tangent lines, identifying special points, and understanding local shape — is a core skill in calculus and lays groundwork for multivariable calculus and differential geometry.
- Set Up the Implicit Equation
- Differentiate Implicitly
- Find Horizontal Tangents
- Find Vertical Tangents
- Identify Singular Points
- Compute the Second Derivative for Concavity
Set Up the Implicit Equation
Write the relation as F(x, y) = 0. For example, for x² + y² = 1 we have F(x, y) = x² + y² − 1 = 0. Recognize that y implicitly depends on x wherever the curve is smooth (by the Implicit Function Theorem, this works when ∂F/∂y ≠ 0).
Differentiate Implicitly
Differentiate both sides with respect to x. Every term involving y requires the chain rule: d/dx[g(y)] = g'(y)·(dy/dx). Collect all dy/dx terms on one side and solve: dy/dx = −(∂F/∂x)/(∂F/∂y). For x² + y² = 1, differentiating gives 2x + 2y(dy/dx) = 0, so dy/dx = −x/y.
Find Horizontal Tangents
Horizontal tangents occur where dy/dx = 0, which requires ∂F/∂x = 0 (the numerator is zero) while ∂F/∂y ≠ 0. For x² + y² = 1, dy/dx = −x/y = 0 means x = 0. Substituting back gives y = ±1, so horizontal tangents are at (0, 1) and (0, −1).
Find Vertical Tangents
Vertical tangents occur where dy/dx is undefined, i.e., where ∂F/∂y = 0 while ∂F/∂x ≠ 0. For the circle, ∂F/∂y = 2y = 0 means y = 0, giving the points (±1, 0). The slope is undefined (vertical tangent) at these points.
Identify Singular Points
A singular point (also called a node or cusp) occurs where both ∂F/∂x = 0 and ∂F/∂y = 0 simultaneously. At such points the Implicit Function Theorem fails, meaning the curve may cross itself, have a cusp, or form an isolated point. For example, at the origin of the folium x³ + y³ − 3xy = 0, both partial derivatives vanish, creating a node where the curve crosses itself.
Compute the Second Derivative for Concavity
Differentiate dy/dx implicitly a second time to find d²y/dx². Substitute known expressions to simplify. For x² + y² = 1 with dy/dx = −x/y, differentiating again gives d²y/dx² = −(y − x(dy/dx))/y² = −(y − x(−x/y))/y² = −(y² + x²)/y³ = −1/y³. Since x² + y² = 1 on the curve. Where y > 0 (upper semicircle), d²y/dx² = −1/y³ < 0, confirming the curve is concave down.
The Key Equation
- F(x,y) — Implicit function (dimensionless): The expression set equal to zero that defines the curve; e.g., F(x,y) = x² + y² − 1.
- F_x = \partial F/\partial x — Partial derivative with respect to x (dimensionless): The rate of change of F treating y as constant; used as the numerator (with sign change) in the slope formula.
- F_y = \partial F/\partial y — Partial derivative with respect to y (dimensionless): The rate of change of F treating x as constant; appears in the denominator. When F_y = 0, the curve has a vertical tangent or singular point.
- dy/dx — Slope of tangent line (dimensionless): The instantaneous slope of the implicit curve at a given smooth point (x, y).
Worked Example
Problem. Consider the lemniscate-like curve x³ + y³ = 6xy (folium of Descartes). (a) Find dy/dx using implicit differentiation. (b) Find the equation of the tangent line at the point (3, 3). (c) Determine whether (3, 3) is a horizontal or vertical tangent, or neither.
Answer. The tangent line at (3, 3) is y = −x + 6, with slope −1. The tangent is oblique (neither horizontal nor vertical).
Origins · 1 minRené Descartes and how this was discovered
Descartes invented the Cartesian coordinate system, making it possible to describe geometric curves algebraically. His 1637 work La Géométrie introduced the use of equations to define curves, which is the foundation of implicit curve analysis. The folium of Descartes — x³ + y³ = 3xy — is named after him and was studied as a challenge to the calculus methods being developed in his era. Descartes reportedly spent most mornings lying in bed thinking — a habit he kept from childhood. He credited much of his philosophical and mathematical insight to these long, uninterrupted morning reflections.
Related Concepts
- Implicit Differentiation — Implicit differentiation is the primary technique used to find dy/dx for implicitly defined curves; it underpins all slope and tangent analysis on implicit relations.
- Chain Rule — The chain rule is essential when differentiating terms involving y with respect to x in implicit differentiation, producing the dy/dx factor automatically.
- Tangent and Normal Lines — Once dy/dx is found for an implicit curve, writing equations of tangent and normal lines follows the same point-slope procedure as for explicit functions.
- Concavity and the Second Derivative — Analyzing concavity of implicit curves requires computing d²y/dx² by differentiating dy/dx implicitly a second time, connecting to the standard second-derivative test.
- Related Rates — Related rates problems use implicit differentiation with respect to time — the same differentiation technique applied to implicit relations, but where x and y both depend on t.