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

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  1. Chapter 5 Class 12 Continuity and Differentiability
  2. Serial order wise

Transcript

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 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 . logโก๐‘ฅ/logโก๐‘ฅ 1/๐‘ฆ . ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ = 1/๐‘ฅ logโกใ€– (logโก๐‘ฅ )ใ€— + 1/๐‘ฅ 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 is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 8 years. He provides courses for Maths and Science at Teachoo. You can check his NCERT Solutions from Class 6 to 12.