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4PM1/01R Jan 2023, Question 6: The Product and Quotient Rules

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

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The question, in short

For the curve y = e^(x² + 1)/(x² + 1), differentiate and determine the constant K, then find the tangent line at x = −1 in simplified form.

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

Part (a) · 5 marks

Differentiate the given quotient involving e^(x² + 1) and x² + 1, expressing the result in the stated form and state the value of K.

  1. 1Quotient rule: dy/dx = [(x² + 1)·2xe^(x² + 1) − e^(x² + 1)·2x]/(x² + 1)²
  2. 2Take out 2xe^(x² + 1): = 2xe^(x² + 1)(x² + 1 − 1)/(x² + 1)²
  3. 3= 2x³e^(x² + 1)/(x² + 1)², so K = 2

Answerdy/dx = 2x³e^(x² + 1)/(x² + 1)² shown, K = 2

Part (b) · 5 marks

At the point where x = −1, work out the tangent line's equation and simplify it.

  1. 1At x = −1: dy/dx = 2(−1)e²/4 = −e²/2
  2. 2y = e²/2
  3. 3y − e²/2 = −(e²/2)(x + 1)
  4. 4y = −e²x/2

Answery = −e²x/2

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