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Grade 12 Math Lab: Analysis & Inference FREE

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Implicit Differentiation

65 min Grade 12 Math Lab: Analysis & Inference

Learning Objectives

  • Differentiate implicit equations
  • Find dy/dx without solving for y
  • Apply to curves

Key Concepts

Implicit Function

Equation where y is not isolated (e.g., x² + y² = 25)

Chain Rule Application

Differentiate y terms with dy/dx factor

Theory

**When y is not isolated:** Differentiate both sides with respect to x. Every time you differentiate a y-term, multiply by dy/dx. **Steps:** 1. Differentiate each term (chain rule for y) 2. Collect dy/dx terms on one side 3. Factor out dy/dx 4. Solve for dy/dx **Example:** x² + y² = 25 2x + 2y(dy/dx) = 0 → dy/dx = -x/y

Examples

Find dy/dx for x³ + y³ = 6xy

Solution: 3x² + 3y²(dy/dx) = 6y + 6x(dy/dx) → dy/dx = (6y - 3x²)/(3y² - 6x)