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4PM1/01 Nov 2023, Question 8: The Product and Quotient Rules

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

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The question, in short

Differentiate y = 2e^(3x+1)/(5x²) and give dy/dx in a specified prime-number form, then estimate the percentage change in y when x increases by 2%.

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

Part (a) · 5 marks

Find dy/dx of y = 2e^(3x+1)/(5x²), expressing it as Ae^(3x+1)(Bx − A)/(Cx³) with A, B, C prime.

  1. 1Quotient rule: dy/dx = [5x² × 6e^(3x+1) − 10x × 2e^(3x+1)]/(5x²)²
  2. 2= xe^(3x+1)(30x − 20)/(25x⁴)
  3. 3= 10e^(3x+1)(3x − 2)/(25x³)
  4. 4= 2e^(3x+1)(3x − 2)/(5x³)

Answerdy/dx = 2e^(3x+1)(3x − 2)/(5x³)

Part (b) · 3 marks

Using part (a), estimate the percentage change in y for a 2% increase in x, giving it as (Px − Q) with integers P, Q.

  1. 1δy ≈ (dy/dx)δx with δx = 0.02x
  2. 2(δy/y) × 100 = [2e^(3x+1)(3x − 2)/(5x³)] × 0.02x × 100/[2e^(3x+1)/(5x²)]
  3. 3= (3x − 2)/x × 0.02x × 100
  4. 4= 6x − 4

Answer(6x − 4) %

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