Check Full Chapter Explained - Continuity and Differentiability - Continuity and Differentiability Class 12

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

Transcript

Ex 5.5, 13 Find ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ of the functions in, ๐‘ฆ^๐‘ฅ = ๐‘ฅ^๐‘ฆ Given, ๐‘ฆ^๐‘ฅ = ๐‘ฅ^๐‘ฆ Taking log both sides log (๐‘ฆ^๐‘ฅ ) = log (๐‘ฅ^๐‘ฆ ) ๐‘ฅ . log ๐‘ฆ=๐‘ฆ.logโก๐‘ฅ Differentiating both sides ๐‘ค.๐‘Ÿ.๐‘ก.๐‘ฅ. (๐‘‘(๐‘ฅ . log ๐‘ฆ))/๐‘‘๐‘ฅ = ๐‘‘(๐‘ฆ.ใ€– logใ€—โก๐‘ฅ )/๐‘‘๐‘ฅ (As ๐‘™๐‘œ๐‘”โก(๐‘Ž^๐‘ )=๐‘ . ๐‘™๐‘œ๐‘”โก๐‘Ž) Using product Rule As (๐‘ข๐‘ฃ)โ€™ = ๐‘ขโ€™๐‘ฃ + ๐‘ฃโ€™๐‘ข ๐‘‘(๐‘ฅ)/๐‘‘๐‘ฅ . log ๐‘ฆ+ ๐‘‘(logโก๐‘ฆ )/๐‘‘๐‘ฅ . ๐‘ฅ =" " ๐‘‘(๐‘ฆ)/๐‘‘๐‘ฅ " ". log ๐‘ฅ + ๐‘‘(logโก๐‘ฅ )/๐‘‘๐‘ฅ . ๐‘ฆ log ๐‘ฆ+๐‘ฅ . ๐‘‘(logโก๐‘ฆ )/๐‘‘๐‘ฅ . ๐‘ฅ = ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ log ๐‘ฅ + 1/๐‘ฅ . ๐‘ฆ log ๐‘ฆ+๐‘ฅ . ๐‘‘(logโก๐‘ฆ )/๐‘‘๐‘ฅ . ๐‘‘๐‘ฆ/๐‘‘๐‘ฆ = ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ . log ๐‘ฅ + ๐‘ฆ/๐‘ฅ log ๐‘ฆ+๐‘ฅ . ๐‘‘(logโก๐‘ฆ )/๐‘‘๐‘ฆ . ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ = ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ . log ๐‘ฅ + ๐‘ฆ/๐‘ฅ log ๐‘ฆ+๐‘ฅ . 1/๐‘ฆ . ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ = ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ . log ๐‘ฅ + ๐‘ฆ/๐‘ฅ log ๐‘ฆ+ ๐‘ฅ/๐‘ฆ . ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ = ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ . log ๐‘ฅ + ๐‘ฆ/๐‘ฅ ๐‘ฅ/๐‘ฆ . ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ โˆ’ ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ . log ๐‘ฅ = ๐‘ฆ/๐‘ฅ โˆ’ log ๐‘ฆ ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ (๐‘ฅ/๐‘ฆ โˆ’ log ๐‘ฅ) = ๐‘ฆ/๐‘ฅ โˆ’ log ๐‘ฆ ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ ((๐‘ฅ โˆ’ ๐‘ฆ logโก๐‘ฅ)/๐‘ฆ) = (๐‘ฆ โˆ’ ๐‘ฅ logโก๐‘ฆ)/๐‘ฅ ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ = (๐‘ฆ โˆ’ ๐‘ฅ logโก๐‘ฆ)/๐‘ฅ . ๐‘ฆ/(๐‘ฅ โˆ’ ๐‘ฆ logโก๐‘ฅ ) ๐’…๐’š/๐’…๐’™ = ๐’š(๐’š โˆ’ ๐’™ ๐’๐’๐’ˆโก๐’š )/๐’™(๐’™ โˆ’ ๐’š ๐’๐’๐’ˆโก๐’™ )

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Davneet Singh
Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 9 years. He provides courses for Maths and Science at Teachoo.