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Derivative of a logarithm with a variable or non-e base (change of base)

This type of question, asked 5 times: 2 May 2023 S2, 11 May 2023 S2, 11 May 2024 S1, 25 Apr 2025 S1, 16 Apr 2026 S1. All 5 questions below, oldest first.

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  1. Question 1

    2 May 2023 S2

    110
    If f(x)=log⁡x2(log⁡ex)f(x) = \log_{x^2}(\log_e x), then f′(x)f'(x) at x=ex = e is

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  2. Question 2

    11 May 2023 S2

    114
    If f(x)=log⁡x2(log⁡x)f(x) = \log_{x^2}(\log x), then at x=ex=e, f′(x)f'(x) has the value

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  3. Question 3

    11 May 2024 S1

    102
    If y=log⁡sin⁡x(tan⁡x)y = \log_{\sin x}(\tan x), then dydx∣x=π/4\left.\frac{dy}{dx}\right|_{x=\pi/4} has the value

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  4. Question 4

    25 Apr 2025 S1

    149
    If y=log⁡3(log⁡3x)y=\log_{3}\left( \log_{3}x \right) then dy dx\frac{dy}{\text{ }dx} at x=3x= 3 i

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  5. Question 5

    16 Apr 2026 S1

    71
    The derivative of log⁡8(log⁡5x)\log_8(\log_5 x) w. r. t. xx is

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