Lecture Notes: Rational Expression and Difference Quotient

Introduction

• Topic: Difference Quotient
• Context: Useful example in calculus
• Variables:
  • x: A variable
  • h: Another variable

Difference Quotient

• Structure:
  • Numerator: Difference of rational expressions
  • Denominator: h

Solving the Numerator

• Goal: Find common denominator for subtraction of rational expressions
• Method:
  • Common denominator = product of x and x + h
  • Use formula: A/B - C/D = (AD - BC) / BD
    • Helps to multiply and combine numerators correctly
  • Steps:
    • New numerator: 1*x - 1*(x + h)
    • Simplifies to: x - x - h

Simplification

• Result after simplification:
  • -h / (x * (x + h)) all divided by h
• Fraction to multiplication:
  • h in the denominator is h/1
  • Flip to multiply: 1/h

Cancellation and Final Result

• Cancellation:
  • h cancel out in the numerator and denominator
• Final simplified expression:
  • -1 / (x * (x + h))

Practice

• Suggestion:
  • Try solving this on your own to ensure understanding

Conclusion

• Next Topic: Will be covered in the following video