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4PM1/02 Jan 2023, Question 7: The Product and Quotient Rules

Pearson Edexcel International GCSE Further Pure Mathematics 4PM1 · Paper 2 · Calculator · 9 marks

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The question, in short

Given y = e^(2x) cos 2x, use differentiation to establish the stated expressions for the first and second derivatives.

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

Part (a) · 4 marks

Differentiate y = e^(2x) cos 2x using the product rule and rearrange to show dy/dx = 2y − 2e^(2x) sin 2x.

  1. 1Product rule: dy/dx = 2e^(2x) cos 2x + e^(2x)(−2 sin 2x)
  2. 2= 2e^(2x) cos 2x − 2e^(2x) sin 2x
  3. 3= 2y − 2e^(2x) sin 2x

Answerdy/dx = 2y − 2e^(2x) sin 2x shown

Part (b) · 5 marks

Differentiate dy/dx again and use part (a) to show d²y/dx² = 4 dy/dx − 8y.

  1. 1d²y/dx² = 2 dy/dx − 4e^(2x) sin 2x − 4e^(2x) cos 2x
  2. 24e^(2x) cos 2x = 4y
  3. 3From (a): −2e^(2x) sin 2x = dy/dx − 2y, so −4e^(2x) sin 2x = 2 dy/dx − 4y
  4. 4d²y/dx² = 2 dy/dx + 2 dy/dx − 4y − 4y
  5. 5= 4 dy/dx − 8y

Answerd²y/dx² = 4 dy/dx − 8y shown

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