Learn in your speed, with individual attention - Teachoo Maths 1-on-1 Class


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

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.