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Grade 12 · Calculus
Differentiation: The Rules and How Exams Use Them
MEMORANDUM
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  / 20

Answer all questions. Show all your working — marks are awarded for method as well as the final answer. Teacher copy: accept any correct equivalent method.

  1. 1
    Differentiate y = x4. (power rule)
    (2)
    4x3✓✓ (2)
  2. 2
    Differentiate y = 5x3 − 2x2 + 7. (polynomial)
    (2)
    15x2 − 4x✓✓ (2)
  3. 3
    Differentiate y = 2x2 − 3x + 9. (polynomial)
    (2)
    4x − 3✓✓ (2)
  4. 4
    Differentiate y = 3⁄x (write as 3x−1). (rewrite)
    (2)
    −3x−2 = −3⁄x2✓✓ (2)
  5. 5
    Differentiate y = x (write as x^12). (rewrite)
    (2)
    12x^(−12) = 1⁄(2x)✓✓ (2)
  6. 6
    From first principles, differentiate f(x) = 3x. (first principles)
    (2)
    [3(x+h)−3x]/h = 3h/h = 3✓✓ (2)
  7. 7
    Gradient of y = x2 + 2x at x = 1. (gradient)
    (2)
    2x+2 at x=1 = 4✓✓ (2)
  8. 8
    Gradient of y = x3 at x = 2. (gradient)
    (2)
    3x2 at x=2 = 12✓✓ (2)
  9. 9
    Find x where y = x2 − 6x has a turning point. (turning point)
    (2)
    2x−6 = 0 → x = 3✓✓ (2)
  10. 10
    Find the turning points of y = x3 − 12x. (turning point)
    (2)
    3x2−12=0 → x = ±2 → (2,−16) and (−2,16)✓✓ (2)
TOTAL: 20 marks
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