Ex 5.5, 13 - Chapter 5 Class 12 Continuity and Differentiability

Transcript

Ex 5.5, 13 Find 𝑑𝑦﷮𝑑𝑥﷯ of the functions in, 𝑦﷮𝑥﷯ = 𝑥﷮𝑦﷯ Given , 𝑦﷮𝑥﷯ = 𝑥﷮𝑦﷯ Taking log both sides log 𝑦﷮𝑥﷯﷯ = log 𝑥﷮𝑦﷯﷯ 𝑥 . log 𝑦=𝑦. log﷮𝑥﷯ Differentiating both sides 𝑤.𝑟.𝑡.𝑥. 𝑑(𝑥 . log 𝑦)﷮𝑑𝑥﷯ = 𝑑 𝑦. log﷮𝑥﷯﷯﷮𝑑𝑥﷯ 𝑑 𝑥﷯﷮𝑑𝑥﷯ . log 𝑦+ 𝑑 log﷮𝑦﷯﷯﷮𝑑𝑥﷯ . 𝑥 = 𝑑 𝑦﷯﷮𝑑𝑥﷯ . log 𝑥 + 𝑑 log﷮𝑥﷯﷯﷮𝑑𝑥﷯ . 𝑦 log 𝑦+𝑥 . 𝑑 log﷮𝑦﷯﷯﷮𝑑𝑥﷯ . 𝑥 = 𝑑𝑦﷮𝑑𝑥﷯ log 𝑥 + 1﷮𝑥﷯ . 𝑦 log 𝑦+𝑥 . 𝑑 log﷮𝑦﷯﷯﷮𝑑𝑥﷯ . 𝑑𝑦﷮𝑑𝑦﷯ = 𝑑𝑦﷮𝑑𝑥﷯ . log 𝑥 + 𝑦﷮𝑥﷯ log 𝑦+𝑥 . 𝑑 log﷮𝑦﷯﷯﷮𝑑𝑦﷯ . 𝑑𝑦﷮𝑑𝑥﷯ = 𝑑𝑦﷮𝑑𝑥﷯ . log 𝑥 + 𝑦﷮𝑥﷯ log 𝑦+𝑥 . 1﷮𝑦﷯ . 𝑑𝑦﷮𝑑𝑥﷯ = 𝑑𝑦﷮𝑑𝑥﷯ . log 𝑥 + 𝑦﷮𝑥﷯ log 𝑦+ 𝑥﷮𝑦﷯ . 𝑑𝑦﷮𝑑𝑥﷯ = 𝑑𝑦﷮𝑑𝑥﷯ . log 𝑥 + 𝑦﷮𝑥﷯ 𝑥﷮𝑦﷯ . 𝑑𝑦﷮𝑑𝑥﷯ − 𝑑𝑦﷮𝑑𝑥﷯ . log 𝑥 = 𝑦﷮𝑥﷯ − log 𝑦 𝑑𝑦﷮𝑑𝑥﷯ 𝑥﷮𝑦﷯ − log 𝑥﷯ = 𝑦﷮𝑥﷯ − log 𝑦 𝑑𝑦﷮𝑑𝑥﷯ 𝑥 − 𝑦 log﷮𝑥﷯﷮𝑦﷯﷯ = 𝑦 − 𝑥 log﷮𝑦﷯﷮𝑥﷯ 𝑑𝑦﷮𝑑𝑥﷯ = 𝑦 − 𝑥 log﷮𝑦﷯﷮𝑥﷯ . 𝑦﷮𝑥 − 𝑦 log﷮𝑥﷯﷯ 𝒅𝒚﷮𝒅𝒙﷯ = 𝒚 𝒚 − 𝒙 𝒍𝒐𝒈﷮𝒚﷯﷯﷮𝒙 𝒙 − 𝒚 𝒍𝒐𝒈﷮𝒙﷯﷯﷯