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Ex 5.7, 9 - Find second order derivatives of log (log x) - Finding second order derivatives - Normal form

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  1. Chapter 5 Class 12 Continuity and Differentiability
  2. Serial order wise
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Ex 5.7, 9 Find the second order derivatives of the function log﷮ ( log﷮𝑥)﷯﷯ Let y = log﷮ ( log﷮𝑥)﷯﷯ Differentiating 𝑤.𝑟.𝑡.𝑥 . 𝑑𝑦﷮𝑑𝑥﷯ = 𝑑( log﷮ ( log﷮𝑥)﷯﷯)﷮𝑑𝑥﷯ 𝑑𝑦﷮𝑑𝑥﷯ = 1﷮ log﷮𝑥﷯﷯ . 𝑑( log﷮𝑥﷯)﷮𝑑𝑥﷯ 𝑑𝑦﷮𝑑𝑥﷯ = 1﷮ log﷮𝑥﷯﷯ . 1﷮𝑥﷯ 𝑑𝑦﷮𝑑𝑥﷯ = 1﷮ 𝑥 . log﷮𝑥﷯﷯ Again Differentiating 𝑤.𝑟.𝑡.𝑥 𝑑﷮𝑑𝑥﷯ 𝑑𝑦﷮𝑑𝑥﷯﷯ = 𝑑﷮𝑑𝑥﷯ 1﷮ 𝑥 . log﷮𝑥﷯﷯﷯ 𝑑﷮2﷯𝑦﷮𝑑 𝑥﷮2﷯﷯ = 𝑑(1)﷮𝑑𝑥﷯ 𝑥 . log﷮𝑥﷯﷯ − 𝑑 𝑥 . log﷮𝑥﷯﷯﷮𝑑𝑥﷯ . 1 ﷮ 𝑥 . log﷮𝑥﷯﷯﷮2﷯﷯ 𝑑﷮2﷯𝑦﷮𝑑 𝑥﷮2﷯﷯ = 0 . 𝑥 . log﷮𝑥﷯﷯ − 𝑑 𝑥 . log﷮𝑥﷯﷯﷮𝑑𝑥﷯ . 1 ﷮ 𝑥 . log﷮𝑥﷯﷯﷮2﷯﷯ 𝑑﷮2﷯𝑦﷮𝑑 𝑥﷮2﷯﷯ = − 𝑑 𝑥 . log﷮𝑥﷯﷯﷮𝑑𝑥﷯﷮ 𝑥 . log﷮𝑥﷯﷯﷮2﷯﷯ 𝑑﷮2﷯𝑦﷮𝑑 𝑥﷮2﷯﷯ = − 𝑑(𝑥)﷮𝑑𝑥﷯ . log﷮𝑥﷯ + 𝑑( log﷮𝑥)﷯﷮𝑑𝑥﷯ . 𝑥﷯﷮ 𝑥 . log﷮𝑥﷯﷯﷮2﷯﷯ 𝑑﷮2﷯𝑦﷮𝑑 𝑥﷮2﷯﷯ = 1. log﷮𝑥 + 1﷮𝑥﷯ × 𝑥﷯﷯﷮ 𝑥. log﷮𝑥﷯﷯﷮2﷯﷯ 𝑑﷮2﷯𝑦﷮𝑑 𝑥﷮2﷯﷯ = − log﷮𝑥﷯ +1﷯﷮ 𝑥 . log﷮𝑥﷯﷯﷮2﷯﷯ Thus , 𝒅﷮𝟐﷯𝒚﷮𝒅 𝒙﷮𝟐﷯﷯ = − 𝒍𝒐𝒈﷮𝒙﷯ +𝟏﷯﷮ 𝒙 . 𝒍𝒐𝒈﷮𝒙﷯﷯﷮𝟐﷯﷯

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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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