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Derivative after simplifying an inverse-trig expression

This type of question, asked 4 times: 2021 Paper 7, 2021 Paper 21, 14 Jun 2022, 4 Apr 2026 S1. All 4 questions below, oldest first.

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  1. Question 1

    2021 Paper 7

    83
    If f(x)=sin⁡(cos⁡−1(1−22x1+22x))f(x) = \sin\left( \cos^{- 1}\left( \frac{1 -2^{2x}}{1 +2^{2x}} \right) \right) and its first derivative with respect to xx is −balog⁡e2-\frac{b}{a}\log_{e}2 when x=1x = 1, where aa and bb are integers, then the minimum value of ∣a2−b2∣\left| a^{2}-b^{2} \right| is

  2. Question 2

    2021 Paper 21

    75
    If y(x)=cot⁡−1(1+sin⁡x+1−sin⁡x1+sin⁡x−1−sin⁡x),x∈(π2,π)y(x) =\cot^{- 1}\left( \frac{\sqrt{1 + \sin x}+\sqrt{1 - \sin x}}{\sqrt{1 + \sin x}-\sqrt{1 - \sin x}} \right),x\in\left( \frac{\pi}{2},\pi \right), then dydx\frac{dy}{dx} at x=5π6x=\frac{5\pi}{6} is:

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  3. Question 3

    14 Jun 2022

    167
    If y=tan⁡−1(sec⁡x3−tan⁡x3), π2<x3<3π2y =\tan^{- 1}\left( \sec x^{3}- \tan x^{3} \right),\ \frac{\pi}{2}<x^{3}<\frac{3\pi}{2}, then

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  4. Question 4

    4 Apr 2026 S1

    66
    If y=tan⁡−1(3cos⁡x−4sin⁡x4cos⁡x+3sin⁡x)+2tan⁡−1(x1+1−x2)y=\tan^{- 1}\left( \frac{3\cos x- 4\sin x}{4\cos x+ 3\sin x} \right)+ 2\tan^{- 1}\left( \frac{x}{1 +\sqrt{1 -x^{2}}} \right), then dydx\frac{dy}{dx} at x=32x =\frac{\sqrt{3}}{2} is equal to :

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