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4PM1/01 Jun 2020, Question 8: The Product and Quotient Rules

Pearson Edexcel International GCSE Further Pure Mathematics 4PM1 · Paper 1 · Calculator · 10 marks

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The question, in short

Given 2xy + 5y = e^x, differentiate implicitly to find dy/dx, evaluate it at x = 0, and find the normal's equation there in the form px + qy + r = 0.

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Part (a) · 5 marks

Show that differentiating 2xy + 5y = e^x gives dy/dx = y(2x + 3)/(2x + 5).

  1. 1y = e^x/(2x + 5)
  2. 2dy/dx = [e^x(2x + 5) − 2e^x]/(2x + 5)²
  3. 3= e^x/(2x + 5) × (2x + 5 − 2)/(2x + 5)
  4. 4= y(2x + 3)/(2x + 5)

Answerdy/dx = y(2x + 3)/(2x + 5) shown

Part (b) · 2 marks

Find the value of dy/dx at x = 0.

  1. 1x = 0: dy/dx = (e⁰ × 5 − 2e⁰)/5²
  2. 2= 3/25

Answer3/25

Part (c) · 3 marks

Find the normal to 2xy + 5y = e^x at x = 0, giving px + qy + r = 0 with integer p, q, r.

  1. 1x = 0 ⇒ y = e⁰/5 = 1/5
  2. 2Normal gradient = −25/3
  3. 3y − 1/5 = −(25/3)x
  4. 4125x + 15y − 3 = 0

Answer125x + 15y − 3 = 0

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