Example 26 - Find derivative of f(x) = sin-1 x - Class 12 - Examples

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  1. Chapter 5 Class 12 Continuity and Differentiability
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Example 26 Find the derivative of f given by 𝑓 𝑥﷯= 𝑠𝑖𝑛﷮−1﷯ 𝑥 assuming it exists. 𝑓 𝑥﷯= 𝑠𝑖𝑛﷮−1﷯ 𝑥 Let 𝑦= 𝑠𝑖𝑛﷮−1﷯ 𝑥 sin﷮𝑦=𝑥﷯ 𝑥= sin﷮𝑦 ﷯ Differentiating both sides 𝑤.𝑟.𝑡.𝑥 𝑑𝑥﷮𝑑𝑥﷯ = 𝑑 sin﷮𝑦﷯﷯﷮𝑑𝑥﷯ 1 = 𝑑 sin﷮𝑦﷯﷯﷮𝑑𝑥﷯ × 𝑑𝑦﷮𝑑𝑦﷯ 1 = 𝑑 sin﷮𝑦﷯﷯﷮𝑑𝑦﷯ × 𝑑𝑦﷮𝑑𝑥﷯ 1 = cos﷮𝑦﷯ 𝑑𝑦﷮𝑑𝑥﷯ 1﷮ cos﷮𝑦﷯﷯= 𝑑𝑦﷮𝑑𝑥﷯ 𝑑𝑦﷮𝑑𝑥﷯ = 1﷮ 𝒄𝒐𝒔﷮𝒚﷯﷯ 𝑑𝑦﷮𝑑𝑥﷯= 1﷮ ﷮𝟏 − 𝒔𝒊𝒏﷮𝟐﷯ 𝒚﷯﷯ Putting 𝑠𝑖𝑛﷮𝑦=𝑥﷯ 𝑑𝑦﷮𝑑𝑥﷯= 1﷮ ﷮1 − 𝒙﷮𝟐﷯﷯﷯ Hence, 𝒅( 𝒔𝒊𝒏﷮−𝟏﷯ 𝒙 )﷮𝒅𝒙﷯ = 𝟏﷮ ﷮𝟏 − 𝒙﷮𝟐﷯﷯﷯

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