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Misc 7 - Differentiate (log x) log x - Chapter 5 Class 12

Misc  7 - Chapter 5 Class 12 Continuity and Differentiability - Part 2
Misc  7 - Chapter 5 Class 12 Continuity and Differentiability - Part 3

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Transcript

Misc 7 Differentiate w.r.t. x the function, (log⁡𝑥 ) log⁡𝑥, 𝑥>1 Let y = (log⁡𝑥 ) log⁡𝑥 Taking log both sides log⁡𝑦 = log ((log⁡𝑥 ) log⁡𝑥 ) log⁡𝑦 = log⁡𝑥. 〖 log〗⁡〖 (log⁡𝑥 )〗 Differentiating both sides 𝑤.𝑟.𝑡.𝑥. 𝑑(log⁡𝑦 )/𝑑𝑥 = 𝑑(log⁡𝑥. 〖 log〗⁡〖 (log⁡𝑥 )〗 )/𝑑𝑥 (As 𝑙𝑜𝑔⁡(𝑎^𝑏) = 𝑏 𝑙𝑜𝑔⁡𝑎) 𝑑(log⁡𝑦 )/𝑑𝑥 (𝑑𝑦/𝑑𝑦) = 𝑑(log⁡𝑥. 〖 log〗⁡〖 (log⁡𝑥 )〗 )/𝑑𝑥 𝑑(log⁡𝑦 )/𝑑𝑦 (𝑑𝑦/𝑑𝑥) = 𝑑(log⁡𝑥. 〖 log〗⁡〖 (log⁡𝑥 )〗 )/𝑑𝑥 1/𝑦 . 𝑑𝑦/𝑑𝑥 = 𝑑(log⁡𝑥. 〖 log〗⁡〖 (log⁡𝑥 )〗 )/𝑑𝑥 1/𝑦 . 𝑑𝑦/𝑑𝑥 = 𝑑(log⁡𝑥 )/𝑑𝑥 . 〖 log〗⁡〖 (log⁡𝑥 )〗 + 𝑑(〖 log〗⁡〖 (log⁡𝑥 )〗 )/𝑑𝑥 .〖 log〗⁡𝑥 1/𝑦 . 𝑑𝑦/𝑑𝑥 = 1/𝑥 log⁡〖 (log⁡𝑥 )〗 + 1/log⁡𝑥 . 𝑑(log⁡𝑥 )/𝑑𝑥 . log⁡𝑥 1/𝑦 . 𝑑𝑦/𝑑𝑥 = 1/𝑥 log⁡〖 (log⁡𝑥 )〗 + (𝑑 (log⁡𝑥 ))/𝑑𝑥 Using Product rule As (𝑢𝑣)’ = 𝑢’𝑣 + 𝑣’𝑢 where u = log x & v = log (log x) 1/𝑦 . 𝑑𝑦/𝑑𝑥 = 1/𝑥 log⁡〖 (log⁡𝑥 )〗 + 1/𝑥 𝑑𝑦/𝑑𝑥 = 𝑦 ( 1/𝑥 + log⁡〖 (log⁡𝑥 )〗/𝑥) 𝒅𝒚/𝒅𝒙 = (𝐥𝐨𝐠⁡𝒙 )^𝐥𝐨𝐠⁡𝒙 (𝟏/𝒙 + 𝒍𝒐𝒈⁡〖 (𝒍𝒐𝒈⁡𝒙 )〗/𝒙) Hence, 𝑑𝑦/𝑑𝑥 = (log⁡𝑥 )^log⁡𝑥 (1/𝑥 + log⁡〖 (log⁡𝑥 )〗/𝑥)

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Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 13 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.