Lecture: Introduction to Implicit Differentiation

Overview

• Last technique of differentiation in calculus.
• Focus on implicit differentiation.
• Recap of slope formula and derivatives.
• Use of dy/dx notation.

Basic Differentiation Examples

1. 3x²:
   • Using power rule:
   • Derivative: 6x
   • Simplifies to: 6x
2. 2y³:
   • Using power rule:
   • Derivative: 6y²
   • Include dy/dx since differentiating with respect to x: 6y² dy/dx
3. x + 3y:
   • Derivative of x: 1
   • Derivative of 3y: 3 dy/dx
   • Result: 1 + 3 dy/dx
4. Product Rule Example: x * y²:
   • First term: x
   • Second term: y²
   • Derivative: x*2y dy/dx + y²
   • Result: 2xy dy/dx + y²

Implicit Differentiation

• Equations not solved explicitly for y.
• Example: x² - 2y + 4y = 2
• Implicit differentiation avoids solving for y.
• Process keeps dy/dx terms.

Steps for Implicit Differentiation

1. Differentiate both sides with respect to x.
2. Collect all dy/dx terms on one side.
3. Factor out dy/dx.
4. Solve for dy/dx.

Example Problems

Example 1: xy = 1

• Explicitly:
  • Solve for y: y = 1/x
  • Derivative: d/dx (1/x) = -1/x²
• Implicitly:
  • Start: xy = 1
  • Differentiate: x dy/dx + y = 0
  • Solve for dy/dx: dy/dx = -y/x
  • Verify using substitution: consistent with explicit solution.

Example 2: Equation with Multiple Y Terms

• Example: 3y² dy/dx - 2x = 0
• Factor out dy/dx: 3y² dy/dx = 2x
• Solve for dy/dx: dy/dx = 2x / 3y²

Example 3: Product Rule Implicit Differentiation

• Complex example: 3x² + 3y² dy/dx = 6x dy/dx + 6y
• Move dy/dx terms to one side: 3x² - 6y = dy/dx (3y² + 6x)
• Solve for dy/dx: dy/dx = (3x² - 6y)/(3y² + 6x)

Additional Practice

• Quotient rule problems can often be simplified by cross-multiplying before differentiating.
• Homework assigned at end.

*Key Points:

• Always include dy/dx when differentiating y terms with respect to x.
• Group dy/dx terms to solve efficiently.
• Use algebraic manipulation to simplify terms before differentiating.