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4PM1/02 Jun 2019, Question 6: The Product and Quotient Rules

Pearson Edexcel International GCSE Further Pure Mathematics 4PM1 · Paper 2 · Calculator · 8 marks

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The question, in short

Differentiate y = (4x − 3)e^(2x), verify a given identity linking (4x − 3)dy/dx and y, then differentiate sin 5x/(x − 3)².

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Part (ai) · 3 marks

Find dy/dx for y = (4x − 3)e^(2x).

  1. 1Product rule: d/dx(4x − 3) = 4, d/dx(e^(2x)) = 2e^(2x)
  2. 2dy/dx = 4e^(2x) + 2(4x − 3)e^(2x)
  3. 3= (8x − 2)e^(2x)

Answerdy/dx = 4e^(2x) + 2(4x − 3)e^(2x)

Part (aii) · 2 marks

Show that (4x − 3)dy/dx equals (8x − 2)y.

  1. 1dy/dx = (8x − 2)e^(2x)
  2. 2(4x − 3) dy/dx = (8x − 2)(4x − 3)e^(2x)
  3. 3= (8x − 2)y

Answer(4x − 3) dy/dx = (8x − 2)y shown

Part (b) · 3 marks

Differentiate sin 5x divided by (x − 3)² with respect to x.

  1. 1Quotient rule with u = sin 5x, u′ = 5 cos 5x, v = (x − 3)², v′ = 2(x − 3)
  2. 2d/dx = [5 cos 5x (x − 3)² − 2 sin 5x (x − 3)]/(x − 3)⁴
  3. 3(= [5(x − 3) cos 5x − 2 sin 5x]/(x − 3)³)

Answer[5 cos 5x (x − 3)² − 2 sin 5x (x − 3)]/(x − 3)⁴

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