Lecture on Differentiation

Introduction to Differentiation

• Differentiation describes the rate of change of one variable with respect to another.
• The method of deriving an equation gives the gradient of the function, also known as the derivative.
• Derivative notation includes:
  • f'(x)
  • y'
  • dy/dx
• dy/dx means the rate of change of y with respect to x.

Continuous vs Discontinuous Functions

• Continuous functions have no discontinuities, such as holes, asymptotes, or jumps.
• A function does not have a derivative where it does not exist.

Differentiation by First Principles

• Formula: f'(x) = lim(h→0) ((f(x+h) - f(x)) / h)
• Example: Find the derivative of f(x) = x² + 4x + 4 using first principles.
  • Substitute into formula and simplify.
  • Result: f'(x) = 2x + 4

General Rule for Derivatives

• For f(x) = x^n, the derivative is f'(x) = n*x^(n-1).
  • Example: y = x² results in y' = 2x.
• Derivative of a sum: Differentiate each term individually.
  • Example: y = x² + 4x + 4
  • Result: y' = 2x + 4*

Derivative of a Multiple

• If y = kx, the derivative is dy/dx = k.
  • Example: y = 4x, derivative is 4.

Examples of Derivatives

1. y = x³ + 3x² + 5
   • Derivative: y' = 3x² + 6x
2. y = 1/2 x^(1/2)
   • Derivative: y' = 1/(2√x)

Applications of Differentiation

• Differentiation can be used to find the gradient at a certain point on a curve.
  • Substitute x into f'(x) to find gradient at that x.
  • Example: y = x² + x - 6, find gradient at x = 2.
    • Result: Gradient is 5.

Practice Problem

• Given y = 3x⁴ - 2x^(-1/2) + 3x + 1, find y'.
  • Result: y' = 12x³ + x^(-3/2) + 3

Important Notes

• Only use first principles when specifically asked.
• Differentiation by first principles can appear in exams.
• Practice using both methods for different types of differentiation questions.