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TMUA 2017 · Paper 1 · Question 2 of 20

TMUA 2017 Paper 1 Question 2

Differentiation and integration — Second derivative of a squared bracket. Try it first; the answer and a full worked solution are below.

TMUA 2017 · Paper 1Differentiation and integrationSecond derivative of a squared bracket6 options
The function f is given by f(x) = (2x − 12x2)2   (x ≠ 0) What is the value of f''(1)?
  1. A−3
  2. B−1
  3. C5
  4. D17
  5. E29
  6. F80
Show the answer and worked solution
answer · C
  1. A−3
  2. B−1
  3. C5
  4. D17
  5. E29
  6. F80
Expand first rather than using the chain rule twice: f(x) = 4x2 − 2⋅2x⋅12x2 + 14x4 = 4x−2 − 2x−3 + 14 x−4. Then f'(x) = −8x−3 + 6x−4 − x−5 and f''(x) = 24x−4 − 24x−5 + 5x−6. At x = 1 every power is 1, so f''(1) = 24 − 24 + 5 = 5.