Implicit Differentiation

Most functions you encounter early in calculus are written explicitly, meaning y is isolated on one side: for example, y = x² + 3x − 1. However, many important mathematical relationships cannot be (or are not conveniently) written this way. A circle of radius 5, described by x² + y² = 25, is a classic example — it fails the vertical line test and cannot be expressed as a single explicit function. Relations like this are called implicitly defined, and we need a special technique to differentiate them. Implicit differentiation is a method that applies the chain rule systematically to both sides of an equation, treating y as an unknown function of x (even though we never solve for y explicitly). Whenever we differentiate a term containing y, we multiply by dy/dx — because y depends on x, and the chain rule demands that we account for that inner function. This produces an equation involving dy/dx, which we then solve algebraically to obtain the derivative. This technique unlocks a vast family of curves — ellipses, hyperbolas, foliums, lemniscates, and more — that arise naturally in physics, engineering, and pure mathematics. It also underlies related-rates problems, where multiple quantities change with time, and it prepares students for advanced topics such as implicit functions in multivariable calculus. Mastering implicit differentiation deepens your understanding of the chain rule and broadens the class of functions you can differentiate.

Key takeaways
  • Differentiate both sides with respect to x
  • Apply the chain rule to every term containing y
  • Apply the product rule to mixed terms
  • Collect all dy/dx terms on one side
  • Solve algebraically for dy/dx

Differentiate both sides with respect to x

Apply d/dx to every term on both sides of the equation. For terms involving only x, differentiate normally. For example, d/dx(x³) = 3x².

Apply the chain rule to every term containing y

Because y is a function of x, the chain rule requires multiplying by dy/dx whenever y is differentiated. For example, d/dx(y³) = 3y² · (dy/dx), and d/dx(sin y) = cos y · (dy/dx).

Apply the product rule to mixed terms

Terms involving both x and y need the product rule. For example, d/dx(x·y) = x·(dy/dx) + y·1 = x(dy/dx) + y.

Collect all dy/dx terms on one side

Move every term containing dy/dx to the left side and every other term to the right side. Factor out dy/dx from all left-side terms.

Solve algebraically for dy/dx

Divide both sides by the factor multiplying dy/dx. The result is the derivative expressed in terms of both x and y — this is normal and expected. To find the slope at a specific point, substitute the coordinates of that point.

The Key Equation

ddx[F(x, y)]=0  ⟹  dydx=−∂F/∂x∂F/∂y\frac{d}{dx}\bigl[F(x,\,y)\bigr] = 0 \implies \frac{dy}{dx} = -\frac{\partial F/\partial x}{\partial F/\partial y}
  • F(x, y) — Implicit relation (dimensionless): An equation set equal to zero that defines y implicitly as a function of x.
  • \partial F/\partial x — Partial derivative with respect to x (dimensionless): Rate of change of F treating y as constant; computed using ordinary differentiation rules.
  • \partial F/\partial y — Partial derivative with respect to y (dimensionless): Rate of change of F treating x as constant; appears in the denominator of the slope formula.
  • dy/dx — Derivative of y with respect to x (dimensionless): The slope of the implicitly defined curve at any point (x, y) on it.

Worked Example

Problem. Find dy/dx for the curve x² + y² = 25 (a circle of radius 5), and determine the slope of the tangent line at the point (3, 4).

dydx=−xy⇒dydx∣(3, 4)=−34\frac{dy}{dx} = -\frac{x}{y} \quad \Rightarrow \quad \left.\frac{dy}{dx}\right|_{(3,\,4)} = -\frac{3}{4}

Answer. dy/dx = −x/y, and the slope at (3, 4) is −3/4.

Figure
Diagram
Implicit Differentiation: Circle x² + y² = 25xy(3, 4)radiusslope = −3/4tangent line−555−5−5x²+y²=25Each axis tick = 1 unit (scaled). Point (3,4): 1 unit ≈ 20px from origin at (200,170).
Origins · 1 minGottfried Wilhelm Leibniz and how this was discovered
GW
Gottfried Wilhelm Leibniz
1646–1716 · German

Leibniz independently developed calculus concurrently with Isaac Newton and introduced the dy/dx notation that makes implicit differentiation transparent and algebraically manipulable. His notation treats differentials as objects that can be multiplied, divided, and rearranged — exactly the intuition needed for implicit differentiation. The chain rule, which lies at the heart of implicit differentiation, was first systematically described in Leibniz's framework. Leibniz was also a philosopher and diplomat who designed an early mechanical calculator. He coined the word 'function' and laid notation foundations still used in calculus classrooms worldwide over 300 years later.

  • The Chain Rule — Implicit differentiation is a direct application of the chain rule — every y-term is differentiated using the chain rule with dy/dx as the inner derivative.
  • Related Rates — Related rates problems differentiate implicit equations with respect to time, extending the same technique to dynamic situations.
  • The Product Rule — Mixed terms such as x·y in implicit equations require the product rule during differentiation, making it an essential companion technique.
  • Tangent and Normal Lines — Implicit differentiation is the primary tool for finding slopes of tangent and normal lines to implicitly defined curves such as circles and ellipses.