The Product and Quotient Rules, Part 2

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• Misc
  • Date for hour exam 1
    • Moving to Monday, Oct. 6 might be good
    • Test will be limits, limit definition of derivative, basic differentiation rules, product & quotient, trig derivatives
    • Problem set re trig derivatives will be due Friday (Oct. 3), to be graded by Thursday Oct. 9
    • (Mostly) up to you
    • Decision: Monday, Oct. 6
  • Exam format
    • 4 - 6 short-answer questions (like problem sets)
    • 75 minutes
    • Open book, notes, computer; closed person
• Questions?
• Product and Quotient Rules
  • Examples: use product and/or quotient rules to find f′(x) if...
    • f(x) = (x² + 2)(x - 1)
      • f′(x) = (2x)(x-1) + (x²+2)(1)
      • = 2x² - 2x + x² + 2
      • = 3x² - 2x + 2
    • f(x) = (x³ - 4x² + 2)( 1 + 1/x)
      • = (x³ - 4x² + 2)( 1 + x^-1)
      • f′(x) = (x³ - 4x² + 2)(-1/x²) + (3x² - 8x)(1+ 1/x)
      • = ( -x³ + 4x² - 2 ) / x² + (3x² - 8x + 3x - 8)
      • = -x + 4 - 2/x² + 3x² - 5x - 8
      • = 3x² - 6x - 2/x² - 4
    • f(x) = x³/x  (f′(x) should be 2x)
      • u(x) = x³, u′(x) = 3x²
      • v(x) = x, v′(x) = 1
      • f′(x) = ( x 3x² - x³ ) / x²
      • = ( 3x³ - x³ ) / x²
      • = 2x³ / x²
      • = 2x
    • f(x) = (x² + 2)/(x - 1)
    • f(x) = 1 / (4 - x²)
    • f(x) = (6x² + 2x - 1)(x³ - 4x) / 7x²
      • = u(x)v(x) / w(x)
      • With...
        • u(x) = 6x² + 2x - 1, so u′(x) = 12x + 2
        • v(x) = x³ - 4x, so v′(x) = 3x² - 4
        • w(x) = 7x², so w′(x) = 14x
      • Two ways to approach this: as a quotient whose numerator is a product, or as a product in which one term is a quotient
        
      • We decided to go with the first approach, i.e.,...
      • f′(x) = ( (uv)′ w - w′ (uv) ) / w²
      • = ( (uv)′ 7x² - w′ (uv) ) / w²
      • = ( (uv)′ 7x² - 14x (uv) ) / w²
      • = ( (uv)′ 7x² - 14x (uv) ) / 49x⁴
      • = ( (uv)′ x - 2 (uv) ) / 7x³
      • = ( (u′v + v′u) x - 2 (uv) ) / 7x³
      • = ( ( (12x + 2)(x³ - 4x) + (3x² - 4)(6x² + 2x - 1) ) x - 2 (uv) ) / 7x³
      • = ( ( (12x⁴ - 48x² + 2x³ - 8x) + (18x⁴ + 6x³ - 3x² - 24x² - 8x + 4) ) x - 2 (uv) ) / 7x³
      • = ( (30x⁴ + 8x³ - 75x² - 16x + 4) x - 2 (uv) ) / 7x³
      • = ( (30x⁵ + 8x⁴ - 75x³ - 16x² + 4x) - 2 (uv) ) / 7x³
      • = ( (30x⁵ + 8x⁴ - 75x³ - 16x² + 4x) - 2 (6x² + 2x - 1)(x³ - 4x) ) / 7x³
      • = ( (30x⁵ + 8x⁴ - 75x³ - 16x² + 4x) - 2 (6x⁵ - 24x³ + 2x⁴ - 8x² - x³ + 4x) ) / 7x³
      • = ( (30x⁵ + 8x⁴ - 75x³ - 16x² + 4x) - 2 (6x⁵ + 2x⁴ - 25x³ - 8x² + 4x) ) / 7x³
      • = ( (30x⁴ + 8x³ - 75x² - 16x + 4) - 2 (6x⁴ + 2x³ - 25x² - 8x + 4) ) / 7x²
      • = ( (30x⁴ + 8x³ - 75x² - 16x + 4) - (12x⁴ + 4x³ - 50x² - 16x + 8) ) / 7x²
      • = ( (18x⁴ + 4x³ - 25x² - 4) ) / 7x²
• Next
  • Derivatives of trigonometric functions
  • Section 3.5

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