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

Transcript

Misc 11 Differentiate w.r.t. x the function, ๐‘ฅ^(๐‘ฅ^2โˆ’ 3)+(๐‘ฅโˆ’3)๐‘ฅ^2, for ๐‘ฅ > 3 Let ๐‘ฆ=๐‘ฅ^(๐‘ฅ^2โˆ’ 3)+(๐‘ฅโˆ’3)^(๐‘ฅ^2 ) Let ๐‘ข=๐‘ฅ^(๐‘ฅ^2โˆ’ 3) , ๐‘ฃ =(๐‘ฅโˆ’3)^(๐‘ฅ^2 ) โˆด 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^(๐‘ฅ^2โˆ’ 3) Taking log on both sides log ๐‘ข=logโกใ€–๐‘ฅ^(๐‘ฅ^2โˆ’ 3) ใ€— log ๐‘ข=ใ€–(๐‘ฅใ€—^2โˆ’ 3). logโก๐‘ฅ Differentiating ๐‘ค.๐‘Ÿ.๐‘ก.๐‘ฅ. ๐‘‘(logโก๐‘ข )/๐‘‘๐‘ฅ = ๐‘‘(ใ€–(๐‘ฅใ€—^2โˆ’ 3) logโก๐‘ฅ )/๐‘‘๐‘ฅ ๐‘‘(logโก๐‘ข )/๐‘‘๐‘ฅ . ๐‘‘๐‘ข/๐‘‘๐‘ข = ๐‘‘(ใ€–(๐‘ฅใ€—^2 โˆ’ 3) logโก๐‘ฅ )/๐‘‘๐‘ฅ " " (As ๐‘™๐‘œ๐‘”โก(๐‘Ž^๐‘) = ๐‘ ๐‘™๐‘œ๐‘”โก๐‘Ž) ๐‘‘(logโก๐‘ข )/๐‘‘๐‘ข . ๐‘‘๐‘ข/๐‘‘๐‘ฅ = ๐‘‘(ใ€–(๐‘ฅใ€—^2โˆ’ 3) logโก๐‘ฅ )/๐‘‘๐‘ฅ " " 1/๐‘ข . ๐‘‘๐‘ข/๐‘‘๐‘ฅ = ๐‘‘(ใ€–(๐‘ฅใ€—^2โˆ’ 3) logโก๐‘ฅ )/๐‘‘๐‘ฅ 1/๐‘ข . ๐‘‘๐‘ข/๐‘‘๐‘ฅ = (๐‘‘ใ€–(๐‘ฅใ€—^2โˆ’ 3) )/๐‘‘๐‘ฅ . ใ€– logใ€—โก๐‘ฅ + ๐‘‘(logโก๐‘ฅ )/๐‘‘๐‘ฅ . ใ€–(๐‘ฅใ€—^2โˆ’ 3) 1/๐‘ข . ๐‘‘๐‘ข/๐‘‘๐‘ฅ = (2๐‘ฅ โˆ’0) ใ€– logใ€—โก๐‘ฅ + (ใ€–(๐‘ฅใ€—^2โˆ’ 3)" " )/๐‘ฅ 1/๐‘ข . ๐‘‘๐‘ข/๐‘‘๐‘ฅ = 2๐‘ฅ . logโก๐‘ฅ + (๐‘ฅ^2โˆ’ 3)/๐‘ฅ Using product rule in ใ€–(๐‘ฅใ€—^2โˆ’ 3). ๐‘™๐‘œ๐‘”โก๐‘ฅ As (๐‘ข๐‘ฃ)โ€™ = ๐‘ขโ€™๐‘ฃ + ๐‘ฃโ€™๐‘ข ๐‘‘๐‘ข/๐‘‘๐‘ฅ = u (2๐‘ฅ "." logโก๐‘ฅ "+ " (๐‘ฅ^2โˆ’ 3)/๐‘ฅ) ๐‘‘๐‘ข/๐‘‘๐‘ฅ = ๐‘ฅ^(๐‘ฅ^2โˆ’ 3) (2๐‘ฅ "." logโก๐‘ฅ "+ " (๐‘ฅ^2โˆ’ 3)/๐‘ฅ) Calculating ๐’…๐’—/๐’…๐’™ ๐‘ฃ= (๐‘ฅโˆ’3)๐‘ฅ^2 Taking log on both sides log ๐‘ฃ=logโกใ€–(๐‘ฅโˆ’3)๐‘ฅ^2 ใ€— log ๐‘ฃ=ใ€–๐‘ฅ^2 . logใ€—โกใ€– (๐‘ฅโˆ’3)ใ€— Differentiating ๐‘ค.๐‘Ÿ.๐‘ก.๐‘ฅ. ๐‘‘(logโก๐‘ฃ )/๐‘‘๐‘ฅ = (๐‘‘(ใ€–๐‘ฅ^2. logใ€—โกใ€– (๐‘ฅโˆ’3)ใ€— ) )/๐‘‘๐‘ฅ (As logโก(๐‘Ž^๐‘) = ๐‘ logโก๐‘Ž) ๐‘‘(logโก๐‘ฃ )/๐‘‘๐‘ฅ . ๐‘‘๐‘ฃ/๐‘‘๐‘ฃ = (๐‘‘(ใ€–๐‘ฅ^2. logใ€—โกใ€– (๐‘ฅโˆ’3)ใ€— ) )/๐‘‘๐‘ฅ ๐‘‘(logโก๐‘ฃ )/๐‘‘๐‘ฃ . ๐‘‘๐‘ฃ/๐‘‘๐‘ฅ = (๐‘‘(ใ€–๐‘ฅ^2. logใ€—โกใ€– (๐‘ฅโˆ’3)ใ€— ) )/๐‘‘๐‘ฅ 1/๐‘ฃ . ๐‘‘๐‘ฃ/๐‘‘๐‘ฅ = (๐‘‘(ใ€–๐‘ฅ^2. logใ€—โกใ€– (๐‘ฅโˆ’3)ใ€— ) )/๐‘‘๐‘ฅ 1/๐‘ฃ . ๐‘‘๐‘ฃ/๐‘‘๐‘ฅ = ๐‘‘(๐‘ฅ^2 )/๐‘‘๐‘ฅ . log (๐‘ฅโˆ’3) + ๐‘‘(log" " (๐‘ฅ โˆ’ 3))/๐‘‘๐‘ฅ . ๐‘ฅ^2 1/๐‘ฃ . ๐‘‘๐‘ฃ/๐‘‘๐‘ฅ = 2๐‘ฅ . log (๐‘ฅโˆ’3) + 1/((๐‘ฅ โˆ’ 3) ). (๐‘‘(๐‘ฅ โˆ’ 3)" " )/๐‘‘๐‘ฅ . ๐‘ฅ^2 1/๐‘ฃ . ๐‘‘๐‘ฃ/๐‘‘๐‘ฅ = 2๐‘ฅ . log (๐‘ฅโˆ’3) + 1/((๐‘ฅ โˆ’ 3) ) . ๐‘ฅ^2 1/๐‘ฃ . ๐‘‘๐‘ฃ/๐‘‘๐‘ฅ = 2๐‘ฅ. log (๐‘ฅโˆ’3) + ๐‘ฅ^2/(๐‘ฅ โˆ’3) Using product rule (๐‘ข๐‘ฃ)โ€™ = ๐‘ขโ€™๐‘ฃ + ๐‘ฃโ€™๐‘ข ๐‘‘๐‘ฃ/๐‘‘๐‘ฅ = ๐‘ฃ (2๐‘ฅ". " log" " (๐‘ฅโˆ’3)" + " ๐‘ฅ^2/(๐‘ฅ โˆ’3)) ๐‘‘๐‘ฃ/๐‘‘๐‘ฅ = (๐‘ฅโˆ’3)๐‘ฅ^2 (2๐‘ฅ". " log" " (๐‘ฅโˆ’3)" + " ๐‘ฅ^2/(๐‘ฅ โˆ’3)) Now, ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ = ๐‘‘๐‘ข/๐‘‘๐‘ฅ + ๐‘‘๐‘ฃ/๐‘‘๐‘ฅ ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ = ๐‘ฅ^(๐‘ฅ^2โˆ’ 3) (2๐‘ฅ "." logโก๐‘ฅ "+ " (๐‘ฅ^2โˆ’ 3)/๐‘ฅ) + (๐‘ฅโˆ’3)๐‘ฅ^2 (2๐‘ฅ". " log" " (๐‘ฅโˆ’3)" + " ๐‘ฅ^2/(๐‘ฅ โˆ’3)) ๐’…๐’š/๐’…๐’™ = ๐’™^(๐’™^๐Ÿโˆ’ ๐Ÿ‘) ((๐’™^๐Ÿโˆ’ ๐Ÿ‘)/๐’™+๐Ÿ๐’™ ๐ฅ๐จ๐ โก๐’™ ) + (๐’™โˆ’๐Ÿ‘)๐’™^๐Ÿ (๐’™^๐Ÿ/(๐’™ โˆ’๐Ÿ‘)+๐Ÿ๐’™ .๐ฅ๐จ๐ โก(๐’™ โˆ’๐Ÿ‘) )

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Davneet Singh's photo - Teacher, Computer Engineer, Marketer
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.