Ex 5.7, 10 - Find the second order derivatives of the function sin (logx) - Ex 5.7

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  1. Chapter 5 Class 12 Continuity and Differentiability
  2. Serial order wise
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Ex 5.7, 10 Find the second order derivatives of the function sin﷮ ( log﷮𝑥)﷯﷯ Let y = sin﷮ ( log﷮𝑥)﷯﷯ Differentiating 𝑤.𝑟.𝑡.𝑥 . 𝑑𝑦﷮𝑑𝑥﷯ = 𝑑( sin﷮ ( log﷮𝑥)﷯﷯)﷮𝑑𝑥﷯ 𝑑𝑦﷮𝑑𝑥﷯ = cos⁡(log⁡𝑥) . 𝑑( log﷮𝑥)﷯﷮𝑑𝑥﷯ 𝑑𝑦﷮𝑑𝑥﷯ = cos⁡(log⁡𝑥) . 1﷮𝑥﷯ 𝑑𝑦﷮𝑑𝑥﷯ = cos⁡(log⁡𝑥)﷮𝑥﷯ Again Differentiating 𝑤.𝑟.𝑡.𝑥 𝑑﷮𝑑𝑥﷯ 𝑑𝑦﷮𝑑𝑥﷯﷯ = 𝑑﷮𝑑𝑥﷯ cos⁡(log⁡𝑥)﷮𝑥﷯﷯ 𝑑﷮2﷯𝑦﷮𝑑 𝑥﷮2﷯﷯ = 𝑑﷮𝑑𝑥﷯ cos⁡(log⁡𝑥)﷮𝑥﷯﷯ 𝑑﷮2﷯𝑦﷮𝑑 𝑥﷮2﷯﷯ = 𝑑(cos⁡(log⁡𝑥))﷮𝑑𝑥﷯ . 𝑥 − 𝑑 𝑥﷯﷮𝑑𝑥﷯ . cos⁡(log⁡𝑥)﷮ 𝑥﷮2﷯﷯ 𝑑﷮2﷯𝑦﷮𝑑 𝑥﷮2﷯﷯ = − sin ﷮ log﷮𝑥﷯﷯﷯ . 𝑑 log﷮𝑥﷯﷯﷯﷮𝑑𝑥﷯ . 𝑥 −1 . cos⁡(log⁡𝑥)﷮ 𝑥﷮2﷯﷯ 𝑑﷮2﷯𝑦﷮𝑑 𝑥﷮2﷯﷯ = − sin ﷮ log﷮𝑥﷯﷯﷯ . 1﷮𝑥﷯ . 𝑥 − cos ⁡(log⁡𝑥)﷮ 𝑥﷮2﷯﷯ 𝑑﷮2﷯𝑦﷮𝑑 𝑥﷮2﷯﷯ = − sin ﷮ log﷮𝑥﷯﷯﷯ − cos⁡(log⁡𝑥)﷮ 𝑥﷮2﷯﷯ 𝒅﷮𝟐﷯𝒚﷮𝒅 𝒙﷮𝟐﷯﷯ = − 𝒔𝒊𝒏 ﷮ 𝒍𝒐𝒈﷮𝒙﷯﷯﷯ + 𝒄𝒐𝒔⁡(𝒍𝒐𝒈⁡𝒙)﷯﷮ 𝒙﷮𝟐﷯﷯

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