Basic Differentiation Rules

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• Questions?
• Differentiation Rules
  • Hand out problem set
  • First part of section 3.3
    • Formulas for derivatives, proven from limit definition
    • Power rule: d(xⁿ)/dx = nx^n-1
    • Sum rule: d(u+v)/dx = du/dx + dv/dx
    • Constant rule: dc/dx = 0
      • vs dx/dx = 1
      • The “dx” on the bottom tells you what counts as a variable for rules
    • Constant multiple rule: d(cx)/dx = c (but see more general form implicit below)
  • Combining rules?
    • Derivative of x⁷ = 7x⁶ using just the power rule
    • Derivative of 7x³ = 21x² using power rule and constant multiple rule
  • Examples
    • d(5x³ - 2x + 7)/dx
      • = 15x² - 2
    • d(x √x)/dx
      • = d( x x^1/2 )/dx
      • = d( x^3/2 )/dx
      • = 3/2 x^1/2
    • d( x^2/3 )/dx
      • = 2/3 x^-1/3
      • (“Exponent minus one” rule can produce either positive or negative exponents)
    • d( x² / 3√x )/dx
      • = d( x² / 3x^1/2 )/dx
      • = d( x^2-1/2 / 3 )/dx
      • = d( x^3/2 / 3 )/dx
      • = d( 1/3 x^3/2 )/dx
      • = 1/2 x^1/2
  • Suppose f′(x) = 3x² and g′(x) = x³ - 1. What is the derivative of 3f(x) - 2g(x)?
    • Mathod 1: work out f(x) and g(x) then differentiate
      • d( 3 f(x) - 2g(x) )/dx
      • = d( 3 (x³ + n ) - 2(1/4 x⁴ - x + n) )/dx
      • = d( 3x³ + 3n - (1/2 x⁴ - 2x + 2n) )/dx
      • = 9x² - 2x³ + 2
    • Or d( 3 f(x) - 2g(x) )/dx
      • = 3 f′(x) - 2 g′(x)
      • = 3(3x²) - 2(x³ - 1 )
      • = 9x² - 2x³ + 2
    • What can you say about what f(x) is? How about g(x)?
      • f(x) = x³ + n
      • g(x) = 1/4 x⁴ - x + n
      • Examples of antiderivatives
  • Position (s), velocity (v), and acceleration (a)
    • v = ds/dt
    • a = dv/dt
    • Constant acceleration a, work out velocity and position as antiderivatives
    • d ?? / dt = a
      • v = at + C
    • s = 1/2 a t² + Ct + K
• Next
  • A bit more on distance, velocity, acceleration
  • Normals (perpendiculars) to curves

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