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Derivative of a composite function at a point from given values of f and f'

This type of question, asked 9 times: 15 May 2023 S2, 16 May 2023 S1, 16 May 2023 S2, 9 May 2024 S1, 10 May 2024 S2, 20 Apr 2025 S1, 22 Apr 2025 S2, 13 Apr 2026 S1, 18 Apr 2026 S1. All 10 questions below, oldest first.

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  1. Question 1

    15 May 2023 S2

    142
    If f(1)=1, f′(1)=3f(1)=1,\, f'(1)=3, then the derivative of f(f(f(x)))+(f(x))2f(f(f(x)))+(f(x))^2 at x=1x=1 is

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  2. Question 2

    16 May 2023 S1

    114
    If f(1)=1,f′(1)=3f(1)=1, f'(1)=3, then the derivative of f(f(f(x)))+(f(x))2f(f(f(x))) + (f(x))^2 at x=1x=1 is

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  3. Question 3

    16 May 2023 S1

    145
    If f(1)=1,f′(1)=5f(1)=1, f'(1)=5, then the derivative of f(f(f(x)))+(f(x))2f(f(f(x))) + (f(x))^2 at x=1x=1 is

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  4. Question 4

    16 May 2023 S2

    101
    If g(x)=[f(2f(x)+2)]2g(x) = [f(2f(x)+2)]^2 and f(0)=−1f(0) = -1, f′(0)=1f'(0) = 1, then g′(0)g'(0) is

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  5. Question 5

    9 May 2024 S1

    125
    If f(1)=1,f′(1)=3f(1)=1, f'(1)=3, then the derivative of f(f(f(x)))+(f(x))2f(f(f(x)))+(f(x))^2 at x=1x=1 is

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  6. Question 6

    10 May 2024 S2

    119
    Let f be a differentiable function such that f(1)=2f(1)=2 and f′(x)=f(x)f'(x)=f(x), for all x∈Rx\in\mathbb{R}. If h(x)=f(f(x))h(x)=f(f(x)), then h′(1)h'(1) is equal to

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  7. Question 7

    20 Apr 2025 S1

    142
    If f(1)=1,f′(1)=3f(1) = 1,f^{'}(1) = 3, then the derivative of f(f(f(x)))+(f(x))2f(f(f(x))) + (f(x))^{2} at x=1x= 1 is

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  8. Question 8

    22 Apr 2025 S2

    105
    If f(1)=3,f′(1)=2f(1) = 3,f^{'}(1) = 2, then ddx{log⁡[f(ex+2x)]}\frac{d}{dx}\left\{ \log\left\lbrack f\left( e^{x}+ 2x \right) \right\rbrack \right\} at x=0x = 0 is

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  9. Question 9

    13 Apr 2026 S1

    73
    Let f(x)=x−5f(x) = x - 5. If g(x)=[f(4h(x)+3)]2g(x) = [f(4h(x) + 3)]^2 and h(1)=4,h′(1)=−2h(1) = 4, h'(1) = -2, then g′(1)=g'(1) = .....

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  10. Question 10

    18 Apr 2026 S1

    75
    If g(x)=(x2+2x+1)⋅f(x)g(x) = (x^2 + 2x + 1)\cdot f(x) such that f(0)=5f(0) = 5 and lim⁡x→0f(x)−5x=4\lim_{x \to 0}\dfrac{f(x) - 5}{x} = 4 then g′(0)=g'(0) =

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