Ex 5.5, 3

Transcript

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