Ex 5.7, 3 - Find second order derivatives of x cosx - Finding second order derivatives - Normal form

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  1. Chapter 5 Class 12 Continuity and Differentiability
  2. Serial order wise

Transcript

Ex 5.7, 3 Find the second order derivatives of the function 𝑥. cos﷮𝑥﷯ Let y = 𝑥. cos﷮𝑥﷯ Differentiating 𝑤.𝑟.𝑡.𝑥 . 𝑑𝑦﷮𝑑𝑥﷯ = 𝑑(𝑥. cos﷮𝑥﷯)﷮𝑑𝑥﷯ 𝑑𝑦﷮𝑑𝑥﷯ = 𝑑 𝑥﷯﷮𝑑𝑥﷯ . cos﷮𝑥﷯ + 𝑑( cos﷮𝑥)﷯﷮𝑑𝑥﷯ . 𝑥 𝑑𝑦﷮𝑑𝑥﷯ = cos⁡𝑥+(− sin⁡𝑥 ) . 𝑥 𝑑𝑦﷮𝑑𝑥﷯ = cos⁡𝑥 − 𝑥 sin⁡𝑥 Again Differentiating 𝑤.𝑟.𝑡.𝑥 𝑑﷮𝑑𝑥﷯ 𝑑𝑦﷮𝑑𝑥﷯﷯ = 𝑑 (cos⁡𝑥 − 𝑥 sin⁡𝑥)﷮𝑑𝑥﷯ 𝑑﷮2﷯𝑦﷮𝑑 𝑥﷮2﷯﷯ = 𝑑( cos﷮𝑥)﷯﷮𝑑𝑥﷯ − 𝑑( x sin﷮𝑥)﷯﷮𝑑𝑥﷯ Using product rule in 𝑥 𝑠𝑖𝑛⁡𝑥 𝑑﷮2﷯𝑦﷮𝑑 𝑥﷮2﷯﷯ = − sin⁡𝑥 − 𝑑 𝑥﷯﷮𝑑𝑥﷯. sin﷮𝑥﷯+ 𝑑 sin﷮𝑥﷯﷯﷮𝑑𝑥﷯.𝑥﷯ 𝑑﷮2﷯𝑦﷮𝑑 𝑥﷮2﷯﷯ = − sin⁡𝑥 − sin﷮𝑥+ cos﷮𝑥 . 𝑥﷯﷯﷯ 𝑑﷮2﷯𝑦﷮𝑑 𝑥﷮2﷯﷯ = − sin⁡𝑥 − sin﷮𝑥− x cos﷮𝑥 ﷯﷯ 𝒅﷮𝟐﷯𝒚﷮𝒅 𝒙﷮𝟐﷯﷯ = − 𝒙 𝒄𝒐𝒔﷮𝒙﷯−𝟐 𝒔𝒊𝒏﷮𝒙﷯

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Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 8 years. He provides courses for Maths and Science at Teachoo. You can check his NCERT Solutions from Class 6 to 12.