Misc 16 - If cos y = x cos (a + y), prove dy/dx = cos2 (a+y) - Finding derivative of Implicit functions

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  1. Class 12
  2. Important Question for exams Class 12
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Misc 16 If cos⁡𝑦=𝑥 cos⁡(𝑎 + 𝑦), with cos⁡𝑎 ≠ ± 1, prove that 𝑑𝑦﷮𝑑𝑥﷯ = 𝑐𝑜𝑠﷮2﷯(𝑎 + 𝑦)﷮ sin﷮𝑎 ﷯﷯ Given cos﷮𝑦﷯ = 𝑥 cos⁡(𝑎 + 𝑦) cos﷮𝑦﷯﷮cos⁡(𝑎 + 𝑦)﷯ = 𝑥 𝑥 = cos﷮𝑦﷯﷮cos⁡(𝑎 + 𝑦)﷯ Differentiating 𝑤.𝑟.𝑡.𝑥. 𝑑 𝑥﷯﷮𝑑𝑥﷯ = 𝑑﷮𝑑𝑥﷯ cos﷮𝑦﷯﷮ cos﷮ 𝑎 + 𝑦﷯﷯﷯﷯ 1 = 𝑑﷮𝑑𝑥﷯ cos﷮𝑦﷯﷮ cos﷮ 𝑎 + 𝑦﷯﷯﷯﷯ . 𝑑𝑦﷮𝑑𝑦﷯ 1 = 𝑑﷮𝑑𝑦﷯ cos﷮𝑦﷯﷮ cos﷮ 𝑎 + 𝑦﷯﷯﷯﷯ . 𝑑𝑦﷮𝑑𝑥﷯ 1 = 𝑑 cos﷮𝑦﷯﷯﷮𝑑𝑦﷯ . cos﷮ 𝑎 + 𝑦﷯﷯ − 𝑑 cos﷮ 𝑎 + 𝑦﷯﷯﷯﷮𝑑𝑦﷯ . cos﷮𝑦﷯﷮ cos﷮ 𝑎 + 𝑦﷯﷯﷯﷮2﷯﷯﷯ . 𝑑𝑦﷮𝑑𝑥﷯ 1 = − sin﷮𝑦﷯ . cos﷮ 𝑎 + 𝑦﷯﷯ − −sin ﷮ 𝑎 + 𝑦﷯﷯﷯ 𝑑 𝑎 + 𝑦﷯﷮𝑑𝑦﷯ . cos﷮𝑦﷯﷮ cos﷮2﷯ ﷮ 𝑎 + 𝑦﷯﷯﷯﷯ . 𝑑𝑦﷮𝑑𝑥﷯ 1 = − sin﷮𝑦﷯ . cos﷮ 𝑎 + 𝑦﷯﷯ + sin ﷮ 𝑎 + 𝑦﷯﷯ 0 + 1﷯ . cos﷮𝑦﷯﷮ cos﷮2﷯ ﷮ 𝑎 + 𝑦﷯﷯﷯﷯ . 𝑑𝑦﷮𝑑𝑥﷯ 1 = sin﷮ 𝑎 + 𝑦﷯﷯ . cos﷮𝑦﷯ − cos ﷮ 𝑎 + 𝑦﷯﷯ . sin﷮𝑦﷯﷮ cos﷮2﷯ ﷮ 𝑎 + 𝑦﷯﷯﷯﷯ . 𝑑𝑦﷮𝑑𝑥﷯ 1 = sin﷮ 𝑎 + 𝑦﷯ − 𝑦﷯﷯﷮ cos﷮2﷯ ﷮ 𝑎 + 𝑦﷯﷯﷯ . 𝑑𝑦﷮𝑑𝑥﷯ 1 = sin﷮ 𝑎 + 𝑦 − 𝑦﷯﷯﷮ cos﷮2﷯ ﷮ 𝑎 + 𝑦﷯﷯﷯ . 𝑑𝑦﷮𝑑𝑥﷯ 1 = sin﷮ 𝑎﷯﷯﷮ cos﷮2﷯ ﷮ 𝑎 + 𝑦﷯﷯﷯ . 𝑑𝑦﷮𝑑𝑥﷯ 1 = sin﷮ 𝑎 + 𝑦 − 𝑦﷯﷯﷮ cos﷮2﷯ ﷮ 𝑎 + 𝑦﷯﷯﷯ . 𝑑𝑦﷮𝑑𝑥﷯ 1 = sin﷮ 𝑎﷯﷯﷮ cos﷮2﷯ ﷮ 𝑎 + 𝑦﷯﷯﷯ . 𝑑𝑦﷮𝑑𝑥﷯ cos﷮2﷯ ﷮ 𝑎 + 𝑦﷯﷯﷮ sin﷮ 𝑎﷯﷯﷯ = 𝑑𝑦﷮𝑑𝑥﷯ 𝒅𝒚﷮𝒅𝒙﷯ = 𝒄𝒐𝒔﷮𝟐﷯ ﷮ 𝒂 + 𝒚﷯﷯﷮ 𝒔𝒊𝒏﷮ 𝒂﷯﷯﷯

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Davneet Singh
Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 7 years. He provides courses for Mathematics and Science from Class 6 to 12. You can learn personally from here https://www.teachoo.com/premium/maths-and-science-classes/.
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