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0606/22 Feb/Mar 2025, Question 6: The Product and Quotient Rules

Cambridge IGCSE Additional Mathematics 0606 · Paper 22 (Calculator) · Calculator · 8 marks

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The question, in short

Differentiate y = ln(2x² + 1)/(x + 2), then use the derivative for a small change in x and a related rate problem.

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

Part (a) · 3 marks

Find dy/dx for y = ln(2x² + 1)/(x + 2).

  1. 1d/dx ln(2x² + 1) = 4x/(2x² + 1)
  2. 2Quotient rule: dy/dx = [(x + 2) × 4x/(2x² + 1) − ln(2x² + 1) × 1]/(x + 2)²

Answerdy/dx = ((4x/(2x² + 1))(x + 2) − ln(2x² + 1))/(x + 2)²

Part (b) · 2 marks

With x increasing from 2 to 2 + h for small h, find the approximate change in y.

  1. 1At x = 2: dy/dx = ((8/9)(4) − ln 9)/16 = 0.0849
  2. 2δy ≈ (dy/dx) × δx = 0.0849h

Answer0.0849h

Part (c) · 3 marks

Given y decreases at 0.4 units per second, find the rate of change of x when x = 2.

  1. 1dy/dt = −0.4
  2. 2dx/dt = (dy/dt) ÷ (dy/dx) = −0.4 ÷ 0.0849…
  3. 3dx/dt = −4.71 units per second

Answer−4.71

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