Let f be a differentiable function such that f(1) = 2 and f'(x) = f(x) for all x R . If h(x) = f(f(x)) , then h'(1) is equal to: - Mathematics MCQ Mathematics Mock Test - 4 2026 | ExamTest.live

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Let f be a differentiable function such that f (1) = 2 and f^{'} (x) = f (x) for all x \in R . If h (x) = f (f (x)), then h^{'} (1) is equal to:

Solution & Step-by-step Explanation

1. Find f (x) :Given f^{'} (x) = f (x), the solution is f (x) = C e^{x} .Using f (1) = 2 : 2 = C e^{1} \Rightarrow C = 2/ e .So, f (x) = \frac{2}{e} e^{x} = 2 e^{x - 1} .Note that f^{'} (x) = 2 e^{x - 1} as well.2. Find h^{'} (1) : h (x) = f (f (x)) h^{'} (x) = f^{'} (f (x)) \cdot f^{'} (x) h^{'} (1) = f^{'} (f (1)) \cdot f^{'} (1) Substitute f (1) = 2 and f^{'} (1) = 2 : h^{'} (1) = f^{'} (2) \cdot 2 Substitute x = 2 in f^{'} (x) = 2 e^{x - 1} : f^{'} (2) = 2 e^{2 - 1} = 2 e . h^{'} (1) = (2 e) \cdot 2 = 4 e .

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Let f be a differentiable function such that f (1) = 2 and f^{'} (x) = f (x) for all x \in R . If h (x) = f (f (x)), then h^{'} (1) is equal to:

4 e

4 e^{2}

2 e

2 e^{2}

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