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

Transcript

Ex 5.5,6 Differentiate the functions in, (๐‘ฅ+1/๐‘ฅ)^๐‘ฅ+ ๐‘ฅ^((1+1/๐‘ฅ) ) Let ๐‘ฆ= (๐‘ฅ+1/๐‘ฅ)^๐‘ฅ+ ๐‘ฅ^((1 + 1/๐‘ฅ) ) Let ๐‘ข = (๐‘ฅ+1/๐‘ฅ)^๐‘ฅ , ๐‘ฃ = ๐‘ฅ^((1 + 1/๐‘ฅ) ) ๐‘ฆ = ๐‘ข+๐‘ฃ Differentiating both sides ๐‘ค.๐‘Ÿ.๐‘ก.๐‘ฅ. ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ = (๐‘‘ (๐‘ข + ๐‘ฃ))/๐‘‘๐‘ฅ ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ = ๐‘‘๐‘ข/๐‘‘๐‘ฅ + ๐‘‘๐‘ฃ/๐‘‘๐‘ฅ Calculating ๐’…๐’–/๐’…๐’™ ๐‘ข = (๐‘ฅ+1/๐‘ฅ)^๐‘ฅ Taking log both sides logโก๐‘ข = log (๐‘ฅ+1/๐‘ฅ)^๐‘ฅ logโก๐‘ข = ๐‘ฅ log (๐‘ฅ+1/๐‘ฅ)(As ๐‘™๐‘œ๐‘”โก(๐‘Ž^๐‘ )=๐‘ . ๐‘™๐‘œ๐‘”โก๐‘Ž) Differentiating both sides ๐‘ค.๐‘Ÿ.๐‘ก.๐‘ฅ. ๐‘‘(logโก๐‘ข )/๐‘‘๐‘ฅ = (๐‘‘ (๐‘ฅ log" " (๐‘ฅ + 1/๐‘ฅ)))/๐‘‘๐‘ฅ ๐‘‘(logโก๐‘ข )/๐‘‘๐‘ฅ (๐‘‘๐‘ข/๐‘‘๐‘ข) = (๐‘‘ (๐‘ฅ log" " (๐‘ฅ + 1/๐‘ฅ)))/๐‘‘๐‘ฅ ๐‘‘(logโก๐‘ข )/๐‘‘๐‘ข (๐‘‘๐‘ข/๐‘‘๐‘ฅ)" = " (๐‘‘ (๐‘ฅ log" " (๐‘ฅ + 1/๐‘ฅ)))/๐‘‘๐‘ฅ 1/๐‘ข (๐‘‘๐‘ข/๐‘‘๐‘ฅ)" = " (๐‘‘ (๐‘ฅ log" " (๐‘ฅ + 1/๐‘ฅ)))/๐‘‘๐‘ฅ using product rule in ๐‘ฅ ๐‘™๐‘œ๐‘”" " (๐‘ฅ + 1/๐‘ฅ) As (uv)โ€™ = ๐‘‘๐‘ข/๐‘‘๐‘ฅ (v) + ๐‘‘๐‘ฃ/๐‘‘๐‘ฅ (u) 1/๐‘ข (๐‘‘๐‘ข/๐‘‘๐‘ฅ)" = " ๐‘‘(๐‘ฅ)/๐‘‘๐‘ฅ . log (๐‘ฅ + 1/๐‘ฅ) + ๐‘‘(log" " (๐‘ฅ + 1/๐‘ฅ))/๐‘‘๐‘ฅ . ๐‘ฅ 1/๐‘ข (๐‘‘๐‘ข/๐‘‘๐‘ฅ)" =" 1. log (๐‘ฅ + 1/๐‘ฅ) + ((1/(๐‘ฅ + 1/๐‘ฅ)).๐‘‘/๐‘‘๐‘ฅ (๐‘ฅ + 1/๐‘ฅ)) . ๐‘ฅ 1/๐‘ข (๐‘‘๐‘ข/๐‘‘๐‘ฅ)" =" log (๐‘ฅ + 1/๐‘ฅ) + (1/(๐‘ฅ + 1/๐‘ฅ) . (๐‘‘(๐‘ฅ)/๐‘‘๐‘ฅ+(๐‘‘ (1/๐‘ฅ))/๐‘‘๐‘ฅ)) . ๐‘ฅ 1/๐‘ข (๐‘‘๐‘ข/๐‘‘๐‘ฅ)" =" log (๐‘ฅ+1/๐‘ฅ) + (1/(๐‘ฅ + 1/๐‘ฅ) . (1+(โˆ’1)/๐‘ฅ^2 " " )) . ๐‘ฅ 1/๐‘ข (๐‘‘๐‘ข/๐‘‘๐‘ฅ)" =" log (๐‘ฅ+1/๐‘ฅ) + (๐‘ฅ/(๐‘ฅ^2 + 1) . (1โˆ’1/๐‘ฅ^2 " " )) . ๐‘ฅ 1/๐‘ข (๐‘‘๐‘ข/๐‘‘๐‘ฅ)" =" log (๐‘ฅ+1/๐‘ฅ) + (๐‘ฅ/(๐‘ฅ^2 + 1) ((๐‘ฅ^2 โˆ’ 1)/๐‘ฅ^2 ).๐‘ฅ) 1/๐‘ข (๐‘‘๐‘ข/๐‘‘๐‘ฅ)" =" log (๐‘ฅ+1/๐‘ฅ) + (๐‘ฅ/(๐‘ฅ^2 + 1) ((๐‘ฅ^2 โˆ’ 1)/๐‘ฅ^2 ).๐‘ฅ) 1/๐‘ข (๐‘‘๐‘ข/๐‘‘๐‘ฅ)" =" log (๐‘ฅ+1/๐‘ฅ) + (๐‘ฅ^2/๐‘ฅ^2 ((๐‘ฅ^2 โˆ’ 1)/(๐‘ฅ^2 + 1))) 1/๐‘ข (๐‘‘๐‘ข/๐‘‘๐‘ฅ)" =" log (๐‘ฅ+1/๐‘ฅ) + ((๐‘ฅ^2 โˆ’ 1)/(๐‘ฅ^2+ 1)) ๐‘‘๐‘ข/๐‘‘๐‘ฅ "= " ๐‘ข (ใ€–log ใ€—โก(๐‘ฅ+1/๐‘ฅ)+((๐‘ฅ^2 โˆ’ 1)/(๐‘ฅ^2+ 1))) ๐‘‘๐‘ข/๐‘‘๐‘ฅ "=" (๐‘ฅ+1/๐‘ฅ)^๐‘ฅ (ใ€–log ใ€—โก(๐‘ฅ+1/๐‘ฅ)+((๐‘ฅ^2 โˆ’ 1)/(๐‘ฅ^2+ 1))) ๐’…๐’–/๐’…๐’™ "=" (๐’™+๐Ÿ/๐’™)^๐’™ ((๐’™^๐Ÿ โˆ’ ๐Ÿ)/(๐’™^๐Ÿ+ ๐Ÿ)โกใ€– ใ€–๐’๐’๐’ˆ ใ€—โก(๐’™+๐Ÿ/๐’™) ใ€— ) Calculating ๐’…๐’—/๐’…๐’™ ๐‘ฃ = ๐‘ฅ^(1 + 1/๐‘ฅ)" " Taking log both sides log ๐‘ฃ = log ๐‘ฅ^(1 + 1/๐‘ฅ)" " log ๐‘ฃ = (1 + 1/๐‘ฅ)log ๐‘ฅ^" " Differentiating both sides ๐‘ค.๐‘Ÿ.๐‘ก.๐‘ฅ. ๐‘‘(logโก๐‘ฃ )/๐‘‘๐‘ฅ = (๐‘‘ ((1 + 1/๐‘ฅ)" . " log ๐‘ฅ))/๐‘‘๐‘ฅ ๐‘‘(logโก๐‘ฃ )/๐‘‘๐‘ฅ (๐‘‘๐‘ฃ/๐‘‘๐‘ฃ) = (๐‘‘ ((1 + 1/๐‘ฅ)" . " log ๐‘ฅ))/๐‘‘๐‘ฅ ๐‘‘(logโก๐‘ฃ )/๐‘‘๐‘ฃ (๐‘‘๐‘ฃ/๐‘‘๐‘ฅ) = (๐‘‘ ((1 + 1/๐‘ฅ)" . " log ๐‘ฅ))/๐‘‘๐‘ฅ 1/๐‘ฃ (๐‘‘๐‘ฃ/๐‘‘๐‘ฅ) = (๐‘‘ ((1 + 1/๐‘ฅ)" . " log ๐‘ฅ))/๐‘‘๐‘ฅ using product rule in (๐‘ฅ+ 1/๐‘ฅ)" . " ๐‘™๐‘œ๐‘” ๐‘ฅ 1/๐‘ฃ (๐‘‘๐‘ฃ/๐‘‘๐‘ฅ) = ๐‘‘(1 + 1/๐‘ฅ)/๐‘‘๐‘ฅ . logโก๐‘ฅ + ๐‘‘(logโก๐‘ฅ )/๐‘‘๐‘ฅ . (1 + 1/๐‘ฅ) 1/๐‘ฃ (๐‘‘๐‘ฃ/๐‘‘๐‘ฅ) = (๐‘‘(1)/๐‘‘๐‘ฅ+๐‘‘(1/๐‘ฅ)/๐‘‘๐‘ฅ) . logโก๐‘ฅ + 1/๐‘ฅ (1 + 1/๐‘ฅ) 1/๐‘ฃ (๐‘‘๐‘ฃ/๐‘‘๐‘ฅ) = (0+((โˆ’1)/๐‘ฅ^2 )) . logโก๐‘ฅ + 1/๐‘ฅ (1 + 1/๐‘ฅ) 1/๐‘ฃ (๐‘‘๐‘ฃ/๐‘‘๐‘ฅ) = (โˆ’1)/๐‘ฅ^2 . logโก๐‘ฅ + 1/๐‘ฅ (1 + 1/๐‘ฅ) 1/๐‘ฃ (๐‘‘๐‘ฃ/๐‘‘๐‘ฅ) = (โˆ’logโก๐‘ฅ)/๐‘ฅ^2 + 1/๐‘ฅ + 1/๐‘ฅ^2 1/๐‘ฃ (๐‘‘๐‘ฃ/๐‘‘๐‘ฅ) = (โˆ’logโก๐‘ฅ)/๐‘ฅ^2 + 1/๐‘ฅ + 1/๐‘ฅ^2 1/๐‘ฃ (๐‘‘๐‘ฃ/๐‘‘๐‘ฅ) = ((โˆ’logโก๐‘ฅ + ๐‘ฅ + 1)/๐‘ฅ^2 ) ๐‘‘๐‘ฃ/๐‘‘๐‘ฅ = ๐‘ฃ ((โˆ’logโก๐‘ฅ + ๐‘ฅ + 1)/๐‘ฅ^2 ) ๐‘‘๐‘ฃ/๐‘‘๐‘ฅ = ๐‘ฅ^((1 + 1/๐‘ฅ) ) ((๐‘ฅ + 1 โˆ’ logโก๐‘ฅ )/๐‘ฅ^2 ) Now ๐‘‘๐‘ฃ/๐‘‘๐‘ฅ = ๐‘‘๐‘ข/๐‘‘๐‘ฅ + ๐‘‘๐‘ฃ/๐‘‘๐‘ฅ Putting values of ๐‘‘๐‘ข/๐‘‘๐‘ฅ & ๐‘‘๐‘ฃ/๐‘‘๐‘ฅ ๐’…๐’š/๐’…๐’™ = (๐’™+๐Ÿ/๐’™)^๐’™ ((๐’™^๐Ÿ โˆ’ ๐Ÿ)/(๐’™^๐Ÿ+ ๐Ÿ)+๐ฅ๐จ๐ โก(๐’™+ ๐Ÿ/๐’™) ) + ๐’™^((๐Ÿ + ๐Ÿ/๐’™) ) ((๐’™ + ๐Ÿ โˆ’ ๐’๐’๐’ˆโก๐’™ )/๐’™^๐Ÿ )

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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.