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4PM1/02R Jan 2022, Question 3: The Product and Quotient Rules

Pearson Edexcel International GCSE Further Pure Mathematics 4PM1 · Paper 2R · Calculator · 6 marks

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The question, in short

Differentiate two expressions with respect to x: a product of e^(2x) and √(5x − 3), and a quotient x³ ÷ cos 3x.

Summarised in our own words. Use the official paper for the exact wording and any diagrams.

Part (a) · 3 marks

Differentiate e^(2x) √(5x − 3) with respect to x.

  1. 1Product rule: u = e^(2x), v = (5x − 3)^(1/2)
  2. 2du/dx = 2e^(2x), dv/dx = (5/2)(5x − 3)^(−1/2)
  3. 3dy/dx = 2e^(2x)√(5x − 3) + 5e^(2x)/(2√(5x − 3))

Answer2e^(2x)√(5x − 3) + 5e^(2x)/(2√(5x − 3))

Part (b) · 3 marks

Differentiate x³ ÷ cos 3x with respect to x.

  1. 1Quotient rule: u = x³, v = cos 3x
  2. 2du/dx = 3x², dv/dx = −3 sin 3x
  3. 3dy/dx = (3x² cos 3x − x³(−3 sin 3x))/(cos 3x)²
  4. 4= (3x² cos 3x + 3x³ sin 3x)/cos²3x

Answer(3x² cos 3x + 3x³ sin 3x)/(cos 3x)²

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