Slide11.JPG

Slide12.JPG
Slide13.JPG
Slide14.JPG
Slide15.JPG Slide16.JPG

Go Ad-free

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 ) And let ๐‘ข=๐‘ฅ^(๐‘ฅ^2โˆ’ 3) , ๐‘ฃ =(๐‘ฅโˆ’3)^(๐‘ฅ^2 ) Now, ๐’š = ๐’–+๐’— 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 " " 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 + 1/๐‘ฅ ร— ใ€–(๐‘ฅใ€—^2โˆ’ 3) 1/๐‘ข . ๐‘‘๐‘ข/๐‘‘๐‘ฅ = 2๐‘ฅ . logโก๐‘ฅ + (๐‘ฅ^2โˆ’ 3)/๐‘ฅ ๐‘‘๐‘ข/๐‘‘๐‘ฅ = u (2๐‘ฅ "." logโก๐‘ฅ "+ " (๐‘ฅ^2โˆ’ 3)/๐‘ฅ) ๐’…๐’–/๐’…๐’™ = ๐’™^(๐’™^๐Ÿโˆ’ ๐Ÿ‘) (๐Ÿ๐’™ "." ๐’๐’๐’ˆโก๐’™ "+ " (๐’™^๐Ÿโˆ’ ๐Ÿ‘)/๐’™) 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‘ฅ โˆ’ 3)ใ€— ) )/๐‘‘๐‘ฅ 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) ๐‘‘๐‘ฃ/๐‘‘๐‘ฅ = ๐‘ฃ (2๐‘ฅ". " log" " (๐‘ฅโˆ’3)" + " ๐‘ฅ^2/(๐‘ฅ โˆ’3)) ๐’…๐’—/๐’…๐’™ = (๐’™โˆ’๐Ÿ‘)๐’™^๐Ÿ (๐Ÿ๐’™". " ๐ฅ๐จ๐ " " (๐’™โˆ’๐Ÿ‘)" + " ๐’™^๐Ÿ/(๐’™ โˆ’๐Ÿ‘)) Now, ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ = ๐‘‘๐‘ข/๐‘‘๐‘ฅ + ๐‘‘๐‘ฃ/๐‘‘๐‘ฅ = 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‘) )

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 14 years. He provides courses for Maths, Science and Computer Science at Teachoo