Ex 5.5, 2 - Differentiate root (x-1)(x-2)/(x-3)(x-4)(x-5)

Ex 5.5, 2 - Chapter 5 Class 12 Continuity and Differentiability - Part 2
Ex 5.5, 2 - Chapter 5 Class 12 Continuity and Differentiability - Part 3

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Transcript

Ex 5.5, 2 Differentiate the functions in, โˆš(((๐‘ฅ โˆ’ 1)(๐‘ฅ โˆ’ 2))/((๐‘ฅ โˆ’ 3)(๐‘ฅ โˆ’ 4)(๐‘ฅ โˆ’ 5))) Let ๐‘ฆ=โˆš(((๐‘ฅ โˆ’ 1)(๐‘ฅ โˆ’ 2))/((๐‘ฅ โˆ’ 3)(๐‘ฅ โˆ’ 4)(๐‘ฅ โˆ’ 5))) ๐‘ฆ= (((๐‘ฅ โˆ’ 1)(๐‘ฅ โˆ’ 2))/((๐‘ฅ โˆ’ 3)(๐‘ฅ โˆ’ 4)(๐‘ฅ โˆ’ 5)))^(1/2) Taking log both sides logโก๐‘ฆ = log (((๐‘ฅ โˆ’ 1)(๐‘ฅ โˆ’ 2))/((๐‘ฅ โˆ’ 3)(๐‘ฅ โˆ’ 4)(๐‘ฅ โˆ’ 5)))^(1/2) logโก๐‘ฆ = 1/2 log (((๐‘ฅ โˆ’ 1)(๐‘ฅ โˆ’ 2))/((๐‘ฅ โˆ’ 3)(๐‘ฅ โˆ’ 4)(๐‘ฅ โˆ’ 5))) (As ๐‘™๐‘œ๐‘”โก(๐‘Ž^๐‘) = ๐‘ ๐‘™๐‘œ๐‘”โก๐‘Ž) logโก๐‘ฆ = 1/2 [logโกใ€–((๐‘ฅโˆ’1)(๐‘ฅโˆ’2))โˆ’logโก((๐‘ฅโˆ’3)(๐‘ฅโˆ’4)(๐‘ฅโˆ’5)) ใ€— ] logโก๐‘ฆ = 1/2 . [("log " (๐‘ฅ+1)" + log " (๐‘ฅโˆ’2))" โˆ’ " (logโก(๐‘ฅโˆ’3)+logโกใ€–(๐‘ฅโˆ’4)+logโก(๐‘ฅโˆ’5) ใ€— )] logโก๐‘ฆ = 1/2 . ["log " (๐‘ฅ+1)" + log " (๐‘ฅ+2)" โˆ’ " logโก(๐‘ฅโˆ’3)โˆ’logโกใ€–(๐‘ฅโˆ’4)โˆ’logโก(๐‘ฅโˆ’5) ใ€— ] Differentiating both sides ๐‘ค.๐‘Ÿ.๐‘ก.๐‘ฅ. ๐‘‘(logโก๐‘ฆ)/๐‘‘๐‘ฅ = (๐‘‘ (1/2 " ." logโกใ€–(๐‘ฅ + 1)" +" logโกใ€– (๐‘ฅ + 2)" โˆ’ " logโก(๐‘ฅ โˆ’ 3)โˆ’logโกใ€–(๐‘ฅ โˆ’ 4)โˆ’logโก(๐‘ฅ โˆ’ 5) ใ€— ใ€— ใ€— ))/๐‘‘๐‘ฅ ๐‘‘(logโก๐‘ฆ)/๐‘‘๐‘ฅ (๐‘‘๐‘ฆ/๐‘‘๐‘ฆ) = 1/2 (๐‘‘(ใ€–log ใ€—โกใ€–(๐‘ฅ + 1)" +" logโกใ€–(๐‘ฅ + 2)" โˆ’" logโก(๐‘ฅ โˆ’ 3) โˆ’ logโกใ€–(๐‘ฅ โˆ’ 4) โˆ’ logโก(๐‘ฅ โˆ’ 5) ใ€— ใ€— ใ€— )/๐‘‘๐‘ฅ) 1/๐‘ฆ . ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ = 1/2 (1/(๐‘ฅ + 1)+1/(๐‘ฅ + 2)โˆ’1/(๐‘ฅ โˆ’ 3)โˆ’1/(๐‘ฅ โˆ’ 4)โˆ’1/(๐‘ฅ โˆ’ 5)) ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ = 1/2 ๐‘ฆ(1/(๐‘ฅ โˆ’ 1)+1/(๐‘ฅ โˆ’ 2)โˆ’1/(๐‘ฅ โˆ’ 3)โˆ’1/(๐‘ฅ โˆ’ 4)โˆ’1/(๐‘ฅ โˆ’ 5)) ๐’…๐’š/๐’…๐’™ = ๐Ÿ/๐Ÿ โˆš(((๐’™ โˆ’ ๐Ÿ)(๐’™ โˆ’ ๐Ÿ))/((๐’™ โˆ’ ๐Ÿ‘)(๐’™ โˆ’ ๐Ÿ’)(๐’™ โˆ’ ๐Ÿ“))) (๐Ÿ/(๐’™ โˆ’ ๐Ÿ)+๐Ÿ/(๐’™ โˆ’ ๐Ÿ)โˆ’๐Ÿ/(๐’™ โˆ’ ๐Ÿ‘)โˆ’๐Ÿ/(๐’™ โˆ’ ๐Ÿ’)โˆ’๐Ÿ/(๐’™ โˆ’ ๐Ÿ“))

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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.