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Derivative of an infinite nested expression (replace the repeating tail by y)

This type of question, asked 6 times: 10 May 2023 S1, 14 May 2024 S2, 23 Apr 2025 S1, 25 Apr 2025 S2, 11 Apr 2026 S1, 15 May 2026 S1. All 6 questions below, oldest first.

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  1. Question 1

    10 May 2023 S1

    103
    If y=sin⁡x1+cos⁡x⋅1+sin⁡x1+cos⁡x⋯y=\frac{\sin x}{1+\cos x}\cdot\frac{1+\sin x}{1+\cos x}\cdots, then dydx\frac{dy}{dx} is given by

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  2. Question 2

    14 May 2024 S2

    112
    If y=(x−sin⁡x)+(x−sin⁡x)+x−sin⁡x+⋯y = \sqrt{(x-\sin x) + \sqrt{(x-\sin x) + \sqrt{x - \sin x + \cdots}}}, then dydx=\frac{dy}{dx} =

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  3. Question 3

    23 Apr 2025 S1

    139
    If y=x+y+x+y+..................∞y=\sqrt{x+\sqrt{y+\sqrt{x+\sqrt{y+ ..................\infty}}}}, then dy dx=\frac{dy}{\text{ }dx}=

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  4. Question 4

    25 Apr 2025 S2

    143
    If y−y−y−........∞=x+x+x+........∞\sqrt{y-\sqrt{y-\sqrt{y- ........\infty}}}=\sqrt{x+\sqrt{x+\sqrt{x+ ........\infty}}} then dy dx=\frac{dy}{\text{ }dx}=

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  5. Question 5

    11 Apr 2026 S1

    71
    If y=tan⁡x+tan⁡x+tan⁡x+⋯⋯∞y = \sqrt{\tan x + \sqrt{\tan x + \sqrt{\tan x + \cdots\cdots\infty}}}, then (dydx)2\left(\dfrac{dy}{dx}\right)^2 at x=π4x = \dfrac{\pi}{4} is

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  6. Question 6

    15 May 2026 S1

    74
    If y=xxx⋅⋅⋅∞y = x^{x^{x^{\cdot^{\cdot^{\cdot\infty}}}}}, then dydx\dfrac{dy}{dx} is:

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