Lecture Notes: Derivatives of Polynomial Functions

Introduction

• Topic: Derivatives of Polynomial Functions
• Focus: Applying derivative rules to various polynomial expressions.

Key Concepts

• Power Rule: Derivative of x^n is n*x^(n-1).
• Constant Multiple Rule: Derivative of a constant times a function is the constant times the derivative of the function.
• Derivative of Constants: The derivative of a constant term is always 0.*

Example 1: Derivative of a Cubic Polynomial

• Function: f(x) = x³ - 5x² + 7x - 4
• Steps:
  • Derivative of x³: 3x² (using power rule)
  • Derivative of 5x²: 5 * 2x = 10x (constant multiple rule)
  • Derivative of 7x: 7
  • Derivative of constant -4: 0
• Result: f'(x) = 3x² - 10x + 7*

Example 2: Derivative of a Quintic Polynomial

• Function: f(x) = 4x⁵ - 6x³ + 8x² - 9
• Steps:
  • Derivative of x⁵: 5x⁴
  • Derivative of x³: 3x²
  • Derivative of x²: 2x
  • Derivative of constant -9: 0
  • Applying constants: 45 = 20, 63 = 18, 8*2 = 16
• Result: f'(x) = 20x⁴ - 18x² + 16x*

Example 3: Derivative of a Product

• Function: f(x) = 7x(2x - x³)
• Steps:
  • Distribute: 7x * 2x = 14x², 7x * -x³ = -7x⁴
  • Derivative of x²: 2x
  • Derivative of x⁴: 4x³
• Result: f'(x) = 28x - 28x³

Example 4: Applying the Chain Rule

• Function: f(x) = (3x + 2)²
• Steps:
  • Expand: (3x + 2)(3x + 2)
  • Foil: 9x² + 6x + 6x + 4 = 9x² + 12x + 4
  • Derivative of x²: 2x
  • Derivative of x: 1
  • Derivative of constant: 0
• Result: f'(x) = 18x + 12

Example 5: Simplifying Before Derivation

• Function: f(x) = (4x⁵ - 5x⁴ + 2x³)/x²
• Steps:
  • Simplify each term:
    • 4x⁵/x² = 4x³
    • 5x⁴/x² = 5x²
    • 2x³/x² = 2x
  • Derive each term:
    • Derivative of x³: 3x²
    • Derivative of x²: 2x
    • Derivative of x: 1
• Result: f'(x) = 12x² - 10x + 2

Conclusion

• Understanding the process of finding derivatives using power rule, constant multiple rule, and simplifying expressions helps in efficiently determining derivatives of polynomial functions.