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4PM1/01R Jun 2022, Question 5: The Product and Quotient Rules

Pearson Edexcel International GCSE Further Pure Mathematics 4PM1 · Paper 1R · Calculator · 8 marks

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The question, in short

Differentiate two expressions with respect to x: a product involving e^(4x) and (6x + 2)^(3/2), and a quotient sin 3x over (2x − 4)³.

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

Part (a) · 5 marks

Differentiate e^(4x)(6x + 2)^(3/2) and express the result as e^(4x)(√(6x + 2))(Ax + B) with integers A and B.

  1. 1Product rule: 4e^(4x)(6x + 2)^(3/2) + e^(4x) × (3/2)(6x + 2)^(1/2) × 6
  2. 2= 4e^(4x)(6x + 2)^(3/2) + 9e^(4x)(6x + 2)^(1/2)
  3. 3= e^(4x)(6x + 2)^(1/2)(4(6x + 2) + 9)
  4. 4= e^(4x)√(6x + 2)(24x + 17)

Answere^(4x)√(6x + 2)(24x + 17)

Part (b) · 3 marks

Differentiate sin 3x divided by (2x − 4)³ with respect to x.

  1. 1Quotient rule with u = sin 3x, v = (2x − 4)³
  2. 2u′ = 3cos 3x, v′ = 3(2x − 4)² × 2 = 6(2x − 4)²
  3. 3dy/dx = (3cos 3x(2x − 4)³ − 6sin 3x(2x − 4)²)/(2x − 4)⁶

Answer(3cos 3x(2x − 4)³ − 6sin 3x(2x − 4)²)/(2x − 4)⁶

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