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Ex 5.7, 7 - Find second order derivatives of e6x cos 3x - 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, 7 Find the second order derivatives of the function 𝑒﷮6𝑥﷯ cos﷮3𝑥﷯ Let y = 𝑒﷮6𝑥﷯ cos﷮3𝑥﷯ Differentiating 𝑤.𝑟.𝑡.𝑥 . 𝑑𝑦﷮𝑑𝑥﷯ = 𝑑( 𝑒﷮6𝑥﷯ cos﷮3𝑥﷯)﷮𝑑𝑥﷯ 𝑑𝑦﷮𝑑𝑥﷯ = 𝑑 𝑒﷮6𝑥﷯﷯﷮𝑑𝑥﷯ . cos﷮3𝑥﷯+ 𝑑( cos﷮3𝑥﷯)﷮𝑑𝑥﷯ . 𝑒﷮6𝑥﷯ 𝑑𝑦﷮𝑑𝑥﷯ = 𝑒﷮6𝑥﷯ . 𝑑 6𝑥﷯﷮𝑑𝑥﷯ . cos﷮3𝑥﷯ + ( −sin﷮3𝑥﷯) . 𝑑(3𝑥)﷮𝑑𝑥﷯ . 𝑒﷮6𝑥﷯ 𝑑𝑦﷮𝑑𝑥﷯ = 𝑒﷮6𝑥﷯ . 6 . cos﷮3𝑥﷯ − sin﷮3𝑥﷯ . 3 . 𝑒﷮6𝑥﷯ 𝑑𝑦﷮𝑑𝑥﷯ = 3 𝑒﷮6𝑥﷯ (2 cos﷮3𝑥﷯ − sin 3𝑥) Again Differentiating 𝑤.𝑟.𝑡.𝑥 𝑑﷮𝑑𝑥﷯ 𝑑𝑦﷮𝑑𝑥﷯﷯ = 𝑑(3 𝑒﷮6𝑥﷯(2 cos﷮3𝑥﷯ − sin﷮3𝑥﷯) )﷮𝑑𝑥﷯ 𝑑﷮2﷯𝑦﷮𝑑 𝑥﷮2﷯﷯ = 3 𝑑( 𝑒﷮6𝑥﷯(2 cos﷮3𝑥﷯ − sin﷮3𝑥﷯) )﷮𝑑𝑥﷯ 𝑑﷮2﷯𝑦﷮𝑑 𝑥﷮2﷯﷯ = 3 𝑑 𝑒﷮6𝑥﷯﷯﷮𝑑𝑥﷯ .(2 cos﷮3𝑥﷯ − sin 3𝑥) + 𝑑(2 cos﷮3𝑥﷯ − sin﷮3𝑥﷯)﷮𝑑𝑥﷯. 𝑒﷮6𝑥﷯﷯ 𝑑﷮2﷯𝑦﷮𝑑 𝑥﷮2﷯﷯ = 3 6 𝑒﷮6𝑥﷯(2 cos﷮3𝑥﷯ − sin 3𝑥) + (−2 sin﷮3𝑥.3﷯ − 𝑐𝑜𝑠3𝑥.3)e6x﷯ 𝑑﷮2﷯𝑦﷮𝑑 𝑥﷮2﷯﷯ = 3 12 𝑒﷮6𝑥﷯ .cos﷮3𝑥﷯− 6 𝑒﷮6𝑥﷯sin 3𝑥− 6 𝑒﷮6𝑥﷯6 sin 3𝑥−3 𝑒﷮6𝑥﷯ cos﷮3𝑥﷯ ﷯ 𝑑﷮2﷯𝑦﷮𝑑 𝑥﷮2﷯﷯ = 3 9 𝑒﷮6𝑥﷯ cos﷮3𝑥﷯−12 𝑒﷮6𝑥﷯. sin﷮3𝑥﷯﷯ 𝑑﷮2﷯𝑦﷮𝑑 𝑥﷮2﷯﷯ = 9 𝑒﷮6𝑥﷯ 3 cos﷮3𝑥﷯−4 sin﷮3𝑥﷯﷯ Hence , 𝒅﷮𝟐﷯𝒚﷮𝒅 𝒙﷮𝟐﷯﷯ = 𝟗 𝒆﷮𝟔𝒙﷯ 𝟑 𝒄𝒐𝒔﷮𝟑𝒙﷯−𝟒 𝒔𝒊𝒏﷮𝟑𝒙﷯﷯

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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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