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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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Misc 7 Differentiate w.r.t. x the function, (logโก๐‘ฅ ) logโก๐‘ฅ, ๐‘ฅ>1 Let y = (logโก๐‘ฅ ) logโก๐‘ฅ Taking log both sides logโก๐‘ฆ = log ((logโก๐‘ฅ ) logโก๐‘ฅ ) logโก๐‘ฆ = logโก๐‘ฅ. ใ€– logใ€—โกใ€– (logโก๐‘ฅ )ใ€— Differentiating both sides ๐‘ค.๐‘Ÿ.๐‘ก.๐‘ฅ. ๐‘‘(logโก๐‘ฆ )/๐‘‘๐‘ฅ = ๐‘‘(logโก๐‘ฅ. ใ€– logใ€—โกใ€– (logโก๐‘ฅ )ใ€— )/๐‘‘๐‘ฅ (As ๐‘™๐‘œ๐‘”โก(๐‘Ž^๐‘) = ๐‘ ๐‘™๐‘œ๐‘”โก๐‘Ž) ๐‘‘(logโก๐‘ฆ )/๐‘‘๐‘ฅ (๐‘‘๐‘ฆ/๐‘‘๐‘ฆ) = ๐‘‘(logโก๐‘ฅ. ใ€– logใ€—โกใ€– (logโก๐‘ฅ )ใ€— )/๐‘‘๐‘ฅ ๐‘‘(logโก๐‘ฆ )/๐‘‘๐‘ฆ (๐‘‘๐‘ฆ/๐‘‘๐‘ฅ) = ๐‘‘(logโก๐‘ฅ. ใ€– logใ€—โกใ€– (logโก๐‘ฅ )ใ€— )/๐‘‘๐‘ฅ 1/๐‘ฆ . ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ = ๐‘‘(logโก๐‘ฅ. ใ€– logใ€—โกใ€– (logโก๐‘ฅ )ใ€— )/๐‘‘๐‘ฅ 1/๐‘ฆ . ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ = ๐‘‘(logโก๐‘ฅ )/๐‘‘๐‘ฅ . ใ€– logใ€—โกใ€– (logโก๐‘ฅ )ใ€— + ๐‘‘(ใ€– logใ€—โกใ€– (logโก๐‘ฅ )ใ€— )/๐‘‘๐‘ฅ .ใ€– logใ€—โก๐‘ฅ 1/๐‘ฆ . ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ = 1/๐‘ฅ logโกใ€– (logโก๐‘ฅ )ใ€— + 1/logโก๐‘ฅ . ๐‘‘(logโก๐‘ฅ )/๐‘‘๐‘ฅ . logโก๐‘ฅ 1/๐‘ฆ . ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ = 1/๐‘ฅ logโกใ€– (logโก๐‘ฅ )ใ€— + (๐‘‘ (logโก๐‘ฅ ))/๐‘‘๐‘ฅ Using Product rule As (๐‘ข๐‘ฃ)โ€™ = ๐‘ขโ€™๐‘ฃ + ๐‘ฃโ€™๐‘ข where u = log x & v = log (log x) 1/๐‘ฆ . ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ = 1/๐‘ฅ logโกใ€– (logโก๐‘ฅ )ใ€— + 1/๐‘ฅ ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ = ๐‘ฆ ( 1/๐‘ฅ + logโกใ€– (logโก๐‘ฅ )ใ€—/๐‘ฅ) ๐’…๐’š/๐’…๐’™ = (๐ฅ๐จ๐ โก๐’™ )^๐ฅ๐จ๐ โก๐’™ (๐Ÿ/๐’™ + ๐’๐’๐’ˆโกใ€– (๐’๐’๐’ˆโก๐’™ )ใ€—/๐’™) Hence, ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ = (logโก๐‘ฅ )^logโก๐‘ฅ (1/๐‘ฅ + logโกใ€– (logโก๐‘ฅ )ใ€—/๐‘ฅ)

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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.