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If ƒ(1) = 1, ƒ'(1) = 3, then the derivative of ƒ(ƒ(ƒ(x))) + (ƒ(x))2 at x = 1 is :

JEE · Math · previous-year question

  1. A.33correct
  2. B.12
  3. C.9
  4. D.15

Answer

A. 33

Explanation

Given ƒ(1) = 1, ƒ'(1) = 3 Let y = ƒ(ƒ(ƒ(x))) + (ƒ(x))2 On differentiating both sides with respect to x we get, $${{dy} \over {dx}}$$ = ƒ'(ƒ(ƒ(x))).ƒ'(ƒ(x)).ƒ'(x) + 2ƒ(x).ƒ'(x) Now at x = 1, $${{dy} \over {dx}}$$ = ƒ'(ƒ(ƒ(1))).ƒ'(ƒ(1)).ƒ'(1) + 2ƒ(1).ƒ'(1) = ƒ'(ƒ(1)).ƒ'(1).ƒ'(1) + 2.1.ƒ'(1) = ƒ'(1).ƒ'(1).ƒ'(1) + 2.1.ƒ'(1) = 3$$ \times $$3$$ \times $$3 + 2$$ \times $$3 = 33

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