Check sibling questions

  Slide20.JPG

Slide21.JPG
Slide22.JPG

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


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 (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)

Ask a doubt
Davneet Singh's photo - Co-founder, Teachoo

Made by

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.