Lecture Notes on Differentiation

Introduction to Differentiation

• Differentiation determines the instantaneous rate of change of a function.
• Example: Y = 3x + 1
  • Derivative, Dy/Dx = 3
• For polynomials like 3x² + 2x:
  • Apply the power rule: Multiply power by coefficient and subtract one from the power.
  • Derivative: 6x + 2

First Principles of Differentiation

• First principle: Derivative function helps determine gradient at a given point.
• Gradient calculation basics:
  • Gradient (m) = (change in y) / (change in x)
• For function 3x² + 2x:
  • Original function: (f(x + h) - f(x)) / h
  • Substitute into derivative equation to simplify.

Fundamental Differentiation Formula

• Formula: If Y = ax^n, then dY/dx = n * ax^(n-1)

Rule for Polynomials and Rational Functions

• Apply power rule to each term.
• For rational functions like 1/x or 1/(2x²):
  • Rewrite using negative exponents if helpful.
  • Differentiate accordingly.

Product Rule

• Handles product of two functions: f(x)g(x)
• Derivative: f'(x)g(x) + f(x)g'(x)
• Example: y = (4x + 2)(2x² +1) yields:
  • Derivative = 8x² + 4 + 24x + 12

Exponential Functions

• Derivative of e^x is e^x
• For e^(2x): Multiply by 2

Quotient Rule

• For division of functions: u/v
• Derivative: (v(u') - u(v')) / v²
• Example: y = x²/(x + 2)

Chain Rule

• For compositions of functions: works on nested functions.
• Example: y = (2x² + 1)^4
  • Derivative: 16x(2x² + 1)^3

Trigonometric Functions

• Common derivatives: sin(x) = cos(x), cos(x) = -sin(x)
• Apply chain rule where necessary.

Logarithmic and Exponential Differentiation

• Natural log (ln(x)) has a derivative of 1/x.
• Exponential e^x retains its form e^x after differentiation.
• For a^x type functions, multiply by ln(a): Derivative = a^x * ln(a)

Implicit Differentiation

• Differentiate both sides of an equation with respect to x, solve for dy/dx
• Useful when equation can't be easily solved for y.

Tangent and Normal Lines

• Tangent: Line that touches curve at one point, parallel to curve at that point.
  • Equation: Use point-slope form y - y1 = m(x - x1)
• Normal: Perpendicular to tangent at point of tangency.
  • Gradient of normal line: -1/m

Increasing and Decreasing Functions

• A function is increasing if its derivative is positive.
• A function is decreasing if its derivative is negative.

Critical Points

• Points where the derivative is zero or undefined.
• Stationary points: Maxima, minima, points of inflection.
• Critical points help determine intervals of increase and decrease.

Practice Problems

• Differentiate given functions and identify critical points, interpret graph behavior.