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TMUA 2016 · Paper 2 · Question 2 of 20

TMUA 2016 Paper 2 Question 2

Differentiation and integration — Differentiating after rewriting as powers of x. Try it first; the answer and a full worked solution are below.

TMUA 2016 · Paper 2Differentiation and integrationDifferentiating after rewriting as powers of x4 options
f(x) = (x2+5)(2x)4√x3,     x > 0

Which one of the following is equal to f'(x)?

  1. A8x94 + 403x14
  2. B92x54 + 52x−34
  3. C8x94 + 403x−14
  4. D813x134 + 8x54
Show the answer and worked solution
answer · B
  1. A8x94 + 403x14
  2. B92x54 + 52x−34
  3. C8x94 + 403x−14
  4. D813x134 + 8x54
Do not reach for the quotient rule. Multiply out the top first: (x2+5)(2x) = 2x3 + 10x, and 4√x3 = x34, so f(x) = 2x94 + 10x14. Differentiating term by term, f'(x) = 2⋅94x54 + 10⋅14x−34 = 92x54 + 52x−34. Options A and C are what you get if you integrate instead of differentiate, or slip a sign in the index.