Ex 5.3, 14 - Find dy/dx in, y= sin-1 (2x root 1-x2) - CBSE - Finding derivative of Inverse trigonometric functions

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  1. Chapter 5 Class 12 Continuity and Differentiability
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Ex 5.3, 14 Find 𝑑𝑦﷮𝑑𝑥﷯ in, y = sin–1 2𝑥 ﷮1− 𝑥﷮2﷯﷯﷯ , − 1﷮ ﷮2﷯﷯ < x < 1﷮ ﷮2﷯﷯ y = sin–1 2𝑥 ﷮1− 𝑥﷮2﷯﷯﷯ Putting 𝑥 = 𝑠𝑖𝑛﷮𝜃﷯ 𝑦 = sin–1 2 sin﷮𝜃﷯ ﷮1− 𝑠𝑖𝑛﷮2﷯𝜃﷯﷯ 𝑦 = sin–1 ( 2 sin θ ﷮ 𝑐𝑜𝑠﷮2﷯𝜃﷯ ) 𝑦 =sin–1 2 sin θ ﷮ cos﷮𝜃﷯﷯﷯ 𝑦 = sin–1 ( sin﷮2 𝜃)﷯ 𝑦 = 2θ Putting value of θ = sin−1 x 𝑦 = 2 𝑠𝑖𝑛﷮−1﷯ 𝑥 Differentiating both sides 𝑤.𝑟.𝑡.𝑥 . 𝑑(𝑦)﷮𝑑𝑥﷯ = 𝑑 2 sin﷮−1﷯﷮𝑥﷯﷯﷮𝑑𝑥﷯ 𝑑𝑦﷮𝑑𝑥﷯ = 2 𝑑 (𝑠𝑖𝑛﷮−1﷯ 𝑥)﷮𝑑𝑥﷯ 𝑑𝑦﷮𝑑𝑥﷯ = 2 1﷮ ﷮1+ 𝑥﷮2﷯﷯﷯﷯ 𝒅𝒚﷮𝒅𝒙﷯ = 𝟐﷮ ﷮𝟏+ 𝒙﷮𝟐﷯﷯﷯

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