If f(1) = 1 , f'(1) = 3 , then the derivative of f(f(f(x))) + (f(x))^2 at x = 1 is: - Mathematics MCQ Mathematics Mock Test - 8 2026 | ExamTest.live

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1 mark

If f (1) = 1, f^{'} (1) = 3, then the derivative of f (f (f (x))) + (f (x))^{2} at x = 1 is:

Solution & Step-by-step Explanation

Let y = f (f (f (x))) + (f (x))^{2} .Using the chain rule: \frac{d y}{d x} = f^{'} (f (f (x))) \cdot f^{'} (f (x)) \cdot f^{'} (x) + 2 f (x) \cdot f^{'} (x) Evaluate at x = 1 :Since f (1) = 1 : f^{'} (f (f (1))) = f^{'} (f (1)) = f^{'} (1) = 3 f^{'} (f (1)) = f^{'} (1) = 3 f^{'} (1) = 3 Substitute values: \frac{d y}{d x}_{x = 1} = (3 \cdot 3 \cdot 3) + (2 \cdot 1 \cdot 3) = 27 + 6 = 33

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If f (1) = 1, f^{'} (1) = 3, then the derivative of f (f (f (x))) + (f (x))^{2} at x = 1 is:

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33

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15

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