Ex 5.5, 1 - Differentiate cos x . cos 2x . cos 3x - Class 12 - Ex 5.5

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

Transcript

Ex 5.5, 1 Differentiate the functions in, cos⁡𝑥 . cos⁡2𝑥 . cos⁡3𝑥 Let y = cos⁡𝑥 . cos⁡2𝑥 . cos⁡3𝑥 Taking log both sides log⁡𝑦 = log cos﷮𝑥﷯. cos﷮2𝑥﷯. cos﷮3𝑥﷯﷯ log⁡𝑦 = log ⁡(cos⁡𝑥) + log ⁡(2𝑥) + log ⁡(cos⁡3𝑥) Differentiating both sides 𝑤.𝑟.𝑡.𝑥. 𝑑 log﷮𝑦﷯﷯﷮𝑑𝑥﷯ = 𝑑 log ⁡(cos⁡𝑥) + log ⁡(2𝑥) + log ⁡( cos﷮3𝑥﷯)﷯﷮𝑑𝑥﷯ 𝑑 log﷮𝑦﷯﷯﷮𝑑𝑥﷯ 𝑑𝑦﷮𝑑𝑦﷯﷯ = 𝑑 log ⁡(cos⁡𝑥)﷯ ﷮𝑑𝑥﷯ + 𝑑 log ⁡(2𝑥)﷯ ﷮𝑑𝑥﷯ + 𝑑 log ⁡( cos﷮3𝑥﷯)﷯ ﷮𝑑𝑥﷯ 𝑑 log﷮𝑦﷯﷯﷮𝑑𝑦﷯ 𝑑𝑦﷮𝑑𝑥﷯﷯ = 1﷮ cos﷮𝑥﷯﷯ . 𝑑 cos﷮𝑥﷯﷯﷮𝑑𝑥﷯ + 1﷮ cos﷮2𝑥﷯﷯ . 𝑑( cos﷮2𝑥﷯)﷮𝑑𝑥﷯ + 1﷮ cos﷮3𝑥﷯﷯ . 𝑑 cos﷮3𝑥﷯﷯﷮𝑑𝑥﷯ 1﷮𝑦﷯ . 𝑑𝑦﷮𝑑𝑥﷯ = 1﷮ cos﷮𝑥﷯﷯ .(− sin⁡𝑥) + 1﷮ cos﷮2𝑥﷯﷯ .(− sin⁡2𝑥). 𝑑 2𝑥﷯﷮𝑑𝑥﷯ + 1﷮ cos﷮𝑥﷯﷯.(− sin⁡3𝑥). 𝑑 3𝑥﷯﷮𝑑𝑥﷯ 1﷮𝑦﷯ . 𝑑𝑦﷮𝑑𝑥﷯ = − sin﷮𝑥﷯﷮ cos﷮𝑥﷯﷯ − sin﷮2𝑥﷯﷮ cos﷮𝑥﷯﷯ . 2 − sin﷮3𝑥﷯﷮ cos﷮3𝑥﷯﷯. 3 1﷮𝑦﷯ . 𝑑𝑦﷮𝑑𝑥﷯ = − tan﷮𝑥﷯− tan﷮2𝑥﷯. 2 − tan﷮3𝑥﷯. 3 1﷮𝑦﷯ . 𝑑𝑦﷮𝑑𝑥﷯ = − tan﷮𝑥﷯+2 tan﷮2𝑥﷯+3 tan﷮3𝑥﷯﷯ 𝑑𝑦﷮𝑑𝑥﷯ = −𝑦 tan﷮𝑥﷯+2 tan﷮2𝑥﷯+3 tan﷮3𝑥﷯﷯ 𝒅𝒚﷮𝒅𝒙﷯ = − 𝒄𝒐𝒔⁡𝒙 . 𝒄𝒐𝒔⁡𝟐𝒙 . 𝒄𝒐𝒔⁡𝟑𝒙 𝒕𝒂𝒏﷮𝒙﷯+𝟐 𝒕𝒂𝒏﷮𝟐𝒙﷯+𝟑 𝒕𝒂𝒏﷮𝟑𝒙﷯﷯

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