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Example 41 - If y = sin-1 x, show that (1 - x2) d2y/dx2 - Finding second order derivatives- Implicit form

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  1. Chapter 5 Class 12 Continuity and Differentiability
  2. Serial order wise
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Example 41 If y = 𝑠𝑖𝑛﷮−1﷯𝑥, show that (1 – 𝑥2) 𝑑2𝑦﷮𝑑𝑥2﷯ − 𝑥 𝑑𝑦﷮𝑑𝑥﷯ = 0 . We have 𝑦 = 𝑠𝑖𝑛﷮−1﷯𝑥 Differentiating 𝑤.𝑟.𝑡.𝑥 𝑑𝑦﷮𝑑𝑥﷯ = 𝑑 𝑠𝑖𝑛﷮−1﷯𝑥﷯﷮𝑑𝑥﷯ 𝑑𝑦﷮𝑑𝑥﷯ = 1﷮ ﷮ 1 − 𝑥﷮2﷯﷯﷯ 𝒅𝒚﷮𝒅𝒙﷯ = 𝟏−𝒙﷮𝟐﷯﷯﷮ −𝟏﷮ 𝟐﷯﷯ Now, 𝑑𝑦﷮𝑑𝑥﷯ = 1−𝑥﷮2﷯﷯﷮ −1﷮ 2﷯﷯ Again Differentiating 𝑤.𝑟.𝑡.𝑥 𝑑﷮𝑑𝑥﷯ 𝑑𝑦﷮𝑑𝑥﷯﷯ = 𝑑 1 − 𝑥﷮2﷯﷯﷮ −1﷮ 2﷯﷯﷮𝑑𝑥﷯ 𝑑﷮2﷯𝑦﷮ 𝑑𝑥﷮2﷯﷯ = −1﷮ 2﷯ 1−𝑥﷮2﷯﷯﷮ −1﷮ 2﷯ −1﷯ . 𝑑 1 − 𝑥﷮2﷯﷯﷮𝑑𝑥﷯ 𝑑﷮2﷯𝑦﷮ 𝑑𝑥﷮2﷯﷯ = −1﷮ 2﷯ 1−𝑥﷮2﷯﷯﷮ −3﷮2﷯ ﷯. 𝑑(1)﷮𝑑𝑥﷯ − 𝑑( 𝑥﷮2﷯)﷮𝑑𝑥﷯﷯ 𝑑﷮2﷯𝑦﷮ 𝑑𝑥﷮2﷯﷯ = −1﷮ 2﷯ 1−𝑥﷮2﷯﷯﷮ −3﷮2﷯ ﷯. 0−2𝑥﷯ 𝑑﷮2﷯𝑦﷮ 𝑑𝑥﷮2﷯﷯ = −1﷮ 2﷯ 1−𝑥﷮2﷯﷯﷮ −3﷮2﷯ ﷯. −2𝑥﷯ 𝒅﷮𝟐﷯𝒚﷮ 𝒅𝒙﷮𝟐﷯﷯ = −𝒙 𝟏−𝒙﷮𝟐﷯﷯﷮ −𝟑﷮𝟐﷯ ﷯ Now, we need to prove 1−𝑥﷮2﷯﷯ 𝑑﷮2﷯𝑦﷮ 𝑑𝑥﷮2﷯﷯ − 𝑥 . 𝑑𝑦﷮𝑑𝑥﷯ = 0 Solving LHS 1−𝑥﷮2﷯﷯ 𝑑﷮2﷯𝑦﷮ 𝑑𝑥﷮2﷯﷯ − 𝑥 . 𝑑𝑦﷮𝑑𝑥﷯ = 1−𝑥﷮2﷯﷯ . 𝑥 1−𝑥﷮2﷯﷯﷮ −3﷮2﷯ ﷯﷯ − 𝑥 1−𝑥﷮2﷯﷯﷮ −1﷮ 2﷯﷯ = 𝑥 1−𝑥﷮2﷯﷯﷮1+ −3﷮2﷯﷯ ﷯−𝑥 1−𝑥﷮2﷯﷯﷮ −1﷮ 2﷯﷯ = 𝑥 1−𝑥﷮2﷯﷯﷮ −1﷮ 2﷯﷯−𝑥 1−𝑥﷮2﷯﷯﷮ −1﷮ 2﷯﷯ = 0 = RHS Hence proved .

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Davneet Singh
Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He provides courses for Mathematics from Class 9 to 12. You can ask questions here.
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