Implicit differentiation

Math

Definition

A technique for finding dy/dx when y is not explicitly solved as a function of x. You differentiate both sides of the equation with respect to x, applying the chain rule to terms involving y, then solve for dy/dx.

How It Works

  1. Differentiate every term on both sides with respect to x.
  2. Apply the chain rule: when differentiating a term with y, multiply by dy/dx.
  3. Collect all terms containing dy/dx on one side of the equation.
  4. Factor out dy/dx.
  5. Solve for dy/dx by dividing.

Examples

  • For x² + y² = 25: 2x + 2y(dy/dx) = 0, so dy/dx = −x/y
  • Finding the slope of the tangent line to an ellipse at a given point
  • Differentiating xy = 1 to get dy/dx = −y/x
Key Fact

Always apply the chain rule to y terms: d/dx[f(y)] = f'(y) · dy/dx

Related Terms

Chain rule Derivative definition Tangent lines Related rates

Study This Concept

Practice implicit differentiation with free review games in these units:

AP Calculus AB Unit 3: Differentiation: Composite, Implicit