Lecture Notes: Finding Derivatives

Introduction

• Derivative of a Constant: Always zero.
  • Example: Derivative of 5 is 0.
• Definition of Derivative: Function that gives the slope at some x-value.
  • Example: f(x) = 8, slope of straight line is 0.

Basic Differentiation Rules

• Power Rule: Derivative of x^n is n * x^(n-1).
  • Example: Derivative of x^2 is 2x.
• Constant Multiple Rule: Derivative of c*f(x) is c times the derivative of f(x).

Examples

• Derivative of x^3: 3x^2
• Derivative of x^4: 4x^3
• Derivative of x^5: 5x^4

Practice Problems

1. Derivative of 4x^7:
   • Use constant multiple rule.
   • Result: 28x^6.
2. Derivative of 8x^4:
   • Result: 32x^3.
3. Derivative of 5x^6:
   • Result: 30x^5.

Advanced Concepts

• Definition of Derivative: Limit as h approaches 0 of (f(x + h) - f(x)) / h.
• Tangent vs Secant Line:
  • Tangent: Touches curve at one point.
  • Secant: Touches curve at two points.

Polynomials

• Derivative of polynomials: Differentiate each term separately.
  • Example: f(x) = x^3 + 7x^2 - 8x + 6
  • Result: 3x^2 + 14x - 8.

Rational Functions

• Rewrite for easier differentiation.
  • Example: f(x) = 1/x becomes x^-1.

Radical Functions

• Convert to rational exponents before differentiating.

Trigonometric Functions

• Derivatives to know:
  • sin(x) -> cos(x)
  • cos(x) -> -sin(x)
  • tan(x) -> sec^2(x)
  • sec(x) -> sec(x)tan(x)
  • cosec(x) -> -cosec(x)cot(x)
  • cot(x) -> -cosec^2(x)

Product Rule

• Derivative of f*g = f' * g + f * g'

Quotient Rule

• Derivative of f/g = (g * f' - f * g') / g^2

Additional Techniques

• Distribute expressions before differentiating.
• Use chain rule for composite functions (not covered in detail).

Conclusion

• Derivatives provide slope of tangent line at any x-value.
• Different techniques apply depending on the function form (polynomial, rational, trigonometric, etc.).