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Ex 5.3, 12 - Find dy/dx in, y = sin-1 (1-x2/1+x2) - Chapter 5 - Ex 5.3

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  1. Chapter 5 Class 12 Continuity and Differentiability
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Ex 5.3, 12 Find 𝑑𝑦﷮𝑑𝑥﷯ in, y = sin–1 1− 𝑥﷮2﷯﷮ 1+ 𝑥2 ﷯﷯ , 0 < x < 1 y = sin–1 1− 𝑥﷮2﷯﷮ 1+ 𝑥2 ﷯﷯ Putting x = tan θ y = 𝑠𝑖𝑛﷮−1﷯ 1− tan﷮2﷯𝜃﷮1+ tan﷮2﷯ 𝜃﷯﷯ y = 𝑠𝑖𝑛﷮−1﷯ ( cos﷮2 𝜃﷯) 𝑦 = 𝑠𝑖𝑛﷮−1﷯ sin ﷮ 𝜋﷮2﷯ −2 𝜃﷯﷯﷯ 𝑦 = 𝜋﷮2﷯ − 2 𝜃 Putting value of θ = tan−1 x 𝑦 = 𝜋﷮2﷯ − 2 𝑡𝑎𝑛﷮−1﷯ 𝑥 Differentiating both sides 𝑤.𝑟.𝑡.𝑥 . 𝑑(𝑦)﷮𝑑𝑥﷯ = 𝑑 𝜋﷮2﷯ − 2 𝑡𝑎𝑛﷮−1﷯ 𝑥 ﷯﷮𝑑𝑥﷯ 𝑑𝑦﷮𝑑𝑥﷯ = 0 − 2 𝑑 (𝑡𝑎𝑛﷮−1﷯ 𝑥)﷮𝑑𝑥﷯ 𝑑𝑦﷮𝑑𝑥﷯ = − 2 𝑑 (𝑡𝑎𝑛﷮−1﷯ 𝑥)﷮𝑑𝑥﷯ 𝑑𝑦﷮𝑑𝑥﷯ = − 2 1﷮1 + 𝑥﷮2﷯﷯﷯ 𝒅𝒚﷮𝒅𝒙﷯ = −𝟐﷮𝟏 + 𝒙﷮𝟐﷯﷯

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