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TMUA 2017 · Paper 2 · Question 1 of 20

TMUA 2017 Paper 2 Question 1

Differentiation and integration — Differentiating after expanding a quotient. Try it first; the answer and a full worked solution are below.

TMUA 2017 · Paper 2Differentiation and integrationDifferentiating after expanding a quotient6 options
Given that y = (1−3x)22x32 which one of the following is a correct expression for dydx?
  1. A94x−12 + 32x−32 − 34x−52
  2. B94x−12 − 32x−32 + 34x−52
  3. C94x−12 − 32x−32 − 34x−52
  4. D−94x−12 + 32x−32 + 34x−52
  5. E−94x−12 + 32x−32 − 34x−52
  6. F−94x−12 − 32x−32 − 34x−52
Show the answer and worked solution
answer · A
  1. A94x−12 + 32x−32 − 34x−52
  2. B94x−12 − 32x−32 + 34x−52
  3. C94x−12 − 32x−32 − 34x−52
  4. D−94x−12 + 32x−32 + 34x−52
  5. E−94x−12 + 32x−32 − 34x−52
  6. F−94x−12 − 32x−32 − 34x−52
There is no need for the quotient rule: expand the top first and split the fraction into separate powers of x. Since (1−3x)2 = 1 − 6x + 9x2, we get y = 12x−32 − 3x−12 + 92x12. Differentiating term by term gives −34x−52 + 32x−32 + 94x−12. Watch the sign on the middle term: it comes from −3 × (−12) = +32.