Ex 5.7, 5 - Find second order derivatives of x3 logx - 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, 5 Find the second order derivatives of the function 𝑥﷮3﷯ log﷮𝑥﷯ Let y = 𝑥﷮3﷯ log﷮𝑥﷯ Differentiating 𝑤.𝑟.𝑡.𝑥 . 𝑑𝑦﷮𝑑𝑥﷯ = 𝑑( 𝑥﷮3﷯ log﷮𝑥﷯)﷮𝑑𝑥﷯ 𝑑𝑦﷮𝑑𝑥﷯ = 𝑑 𝑥﷮3﷯﷯﷮𝑑𝑥﷯ . log﷮𝑥﷯ + 𝑑( log﷮𝑥﷯)﷮𝑑𝑥﷯ . 𝑥﷮3﷯ 𝑑𝑦﷮𝑑𝑥﷯ = 3𝑥2 . log⁡𝑥 + 1﷮𝑥﷯ . 𝑥﷮3﷯ 𝑑𝑦﷮𝑑𝑥﷯ = 3𝑥2 . log⁡𝑥 + 𝑥﷮2﷯ Again Differentiating 𝑤.𝑟.𝑡.𝑥 𝑑﷮𝑑𝑥﷯ 𝑑𝑦﷮𝑑𝑥﷯﷯ = 𝑑 (3𝑥2 . log⁡𝑥 + 𝑥﷮2﷯)﷮𝑑𝑥﷯ 𝑑﷮2﷯𝑦﷮𝑑 𝑥﷮2﷯﷯ = 𝑑 (3𝑥2 . log⁡𝑥)﷮𝑑𝑥﷯ + 𝑑 ( 𝑥﷮2﷯)﷮𝑑𝑥﷯ 𝑑﷮2﷯𝑦﷮𝑑 𝑥﷮2﷯﷯ = 3 𝑑(𝑥2 . log⁡𝑥)﷮𝑑𝑥﷯ + 2 𝑥 𝑑﷮2﷯𝑦﷮𝑑 𝑥﷮2﷯﷯ = 3 . 𝑑 𝑥﷯﷮2﷯﷮𝑑𝑥﷯ . log﷮𝑥+ 𝑑( log﷮𝑥)﷯﷮𝑑𝑥﷯﷯. 𝑥﷮2﷯﷯ + 2 𝑥 = 3 2𝑥 . log﷮𝑥+ 1﷮𝑥﷯﷯. 𝑥﷮2﷯﷯ + 2 𝑥 = 3 2𝑥 . log﷮𝑥﷯+ 𝑥﷯ + 2 𝑥 = 6 𝑥 log⁡𝑥 + 3𝑥 + 2𝑥 = 6 𝑥 log ⁡𝑥+5𝑥 = x 6 log﷮𝑥+5﷯﷯ = 𝑥 5+6 log﷮𝑥﷯﷯ Hence , 𝒅﷮𝟐﷯𝒚﷮𝒅 𝒙﷮𝟐﷯﷯ = 𝒙 𝟓+𝟔 𝒍𝒐𝒈﷮𝒙﷯﷯

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