3.5

3.5 IMPLICIT DIFFERENTIATION: Application of Chain Rule

Study Tips

• Quiz Preparation: Focus on key derivatives and implicit differentiation techniques

• Homework: Start early to avoid last-minute rush

• Questions: Reach out for help during spring break if needed!

Key Points

• Implicit Functions: Understand the definition and characteristics

• Implicit Differentiation Technique: Steps to apply when y cannot be solved for x

• Inverse Trig Functions Derivatives: Finding derivatives using implicit differentiation

Examples

Example 1: Circle Slope

• Circle Equation: x² + y² = 1

• Rewrite: y = ±√(1 - x²) (upper and lower semi-circles)

• Differentiate Upper Semi-circle:y = √(1 - x²)

  • Derivative: y' = -x / √(1 - x²)

• Differentiate Lower Semi-circle:y = -√(1 - x²)

  • Derivative: y' = x / √(1 - x²)

Example 2: Implicit Differentiation for x² + y² = 1

• Step 1: Differentiate:2x + 2y(dy/dx) = 0

• Step 2: Solve for dy/dx:dy/dx = -x/y

Example 3: Slope at (1, 0)

• Equation: x² + 2y = y² + 1

• Derivative at (1,0): y' = -1

• Tangent Line Equation: y = -x + 1

Applications of Implicit Differentiation

• Derivatives of Inverse Functions:

  • arcsin(x): dy/dx = 1 / √(1 - x²)

  • arccos(x): dy/dx = -1 / √(1 - x²)

  • arctan(x): dy/dx = 1 / (1 + x²)

• Composite Function Example:

  • For f(x) = arctan(x² + 1): Use chain rule!