Miscellaneous
Misc 2
Misc 3
Misc 4
Misc 5 Important
Misc 6 Important
Misc 7 Important
Misc 8
Misc 9 Important
Misc 10
Misc 11 Important You are here
Misc 12
Misc 13 Important
Misc 14 Important
Misc 15 Important
Misc 16 Important
Misc 17 Important
Misc 18
Misc 19
Misc 20
Misc 21
Misc 22 Important
Question 1 Important Deleted for CBSE Board 2025 Exams
Last updated at April 16, 2024 by Teachoo
Misc 11 Differentiate w.r.t. x the function, ๐ฅ^(๐ฅ^2โ 3)+(๐ฅโ3)๐ฅ^2, for ๐ฅ > 3 Let ๐ฆ=๐ฅ^(๐ฅ^2โ 3)+(๐ฅโ3)^(๐ฅ^2 ) And let ๐ข=๐ฅ^(๐ฅ^2โ 3) , ๐ฃ =(๐ฅโ3)^(๐ฅ^2 ) Now, ๐ = ๐+๐ Differentiating both sides ๐ค.๐.๐ก.๐ฅ. ๐๐ฆ/๐๐ฅ = (๐ (๐ข + ๐ฃ))/๐๐ฅ ๐๐ฆ/๐๐ฅ = ๐๐ข/๐๐ฅ + ๐๐ฃ/๐๐ฅ Calculating ๐ ๐/๐ ๐ ๐ข = ๐ฅ^(๐ฅ^2โ 3) Taking log on both sides log ๐ข=logโกใ๐ฅ^(๐ฅ^2โ 3) ใ log ๐ข=ใ(๐ฅใ^2โ 3). logโก๐ฅ Differentiating ๐ค.๐.๐ก.๐ฅ. ๐(logโก๐ข )/๐๐ฅ = ๐(ใ(๐ฅใ^2โ 3) logโก๐ฅ )/๐๐ฅ ๐(logโก๐ข )/๐๐ฅ . ๐๐ข/๐๐ข = ๐(ใ(๐ฅใ^2 โ 3) logโก๐ฅ )/๐๐ฅ " " ๐(logโก๐ข )/๐๐ข . ๐๐ข/๐๐ฅ = ๐(ใ(๐ฅใ^2โ 3) logโก๐ฅ )/๐๐ฅ " " 1/๐ข . ๐๐ข/๐๐ฅ = ๐(ใ(๐ฅใ^2โ 3) logโก๐ฅ )/๐๐ฅ 1/๐ข . ๐๐ข/๐๐ฅ = (๐ใ(๐ฅใ^2โ 3) )/๐๐ฅ . ใ logใโก๐ฅ + ๐(logโก๐ฅ )/๐๐ฅ . ใ(๐ฅใ^2โ 3) 1/๐ข . ๐๐ข/๐๐ฅ = (2๐ฅ โ0) ใ logใโก๐ฅ + 1/๐ฅ ร ใ(๐ฅใ^2โ 3) 1/๐ข . ๐๐ข/๐๐ฅ = 2๐ฅ . logโก๐ฅ + (๐ฅ^2โ 3)/๐ฅ ๐๐ข/๐๐ฅ = u (2๐ฅ "." logโก๐ฅ "+ " (๐ฅ^2โ 3)/๐ฅ) ๐ ๐/๐ ๐ = ๐^(๐^๐โ ๐) (๐๐ "." ๐๐๐โก๐ "+ " (๐^๐โ ๐)/๐) Calculating ๐ ๐/๐ ๐ ๐ฃ = (๐ฅโ3)๐ฅ^2 Taking log on both sides log ๐ฃ=logโกใ(๐ฅโ3)^(๐ฅ^2 ) ใ log ๐ฃ=ใ๐ฅ^2 . logใโกใ (๐ฅโ3)ใ Differentiating ๐ค.๐.๐ก.๐ฅ. ๐(logโก๐ฃ )/๐๐ฅ = (๐(ใ๐ฅ^2. logใโกใ (๐ฅ โ 3)ใ ) )/๐๐ฅ ๐(logโก๐ฃ )/๐๐ฅ . ๐๐ฃ/๐๐ฃ = (๐(ใ๐ฅ^2. logใโกใ (๐ฅโ3)ใ ) )/๐๐ฅ ๐(logโก๐ฃ )/๐๐ฃ . ๐๐ฃ/๐๐ฅ = (๐(ใ๐ฅ^2. logใโกใ (๐ฅโ3)ใ ) )/๐๐ฅ 1/๐ฃ . ๐๐ฃ/๐๐ฅ = (๐(ใ๐ฅ^2. logใโกใ (๐ฅโ3)ใ ) )/๐๐ฅ 1/๐ฃ . ๐๐ฃ/๐๐ฅ = ๐(๐ฅ^2 )/๐๐ฅ . log (๐ฅโ3) + ๐(log" " (๐ฅ โ 3))/๐๐ฅ . ๐ฅ^2 1/๐ฃ . ๐๐ฃ/๐๐ฅ = 2๐ฅ . log (๐ฅโ3) + 1/((๐ฅ โ 3) ). (๐(๐ฅ โ 3)" " )/๐๐ฅ . ๐ฅ^2 1/๐ฃ . ๐๐ฃ/๐๐ฅ = 2๐ฅ . log (๐ฅโ3) + 1/((๐ฅ โ 3) ) . ๐ฅ^2 1/๐ฃ . ๐๐ฃ/๐๐ฅ = 2๐ฅ. log (๐ฅโ3) + ๐ฅ^2/(๐ฅ โ3) ๐๐ฃ/๐๐ฅ = ๐ฃ (2๐ฅ". " log" " (๐ฅโ3)" + " ๐ฅ^2/(๐ฅ โ3)) ๐ ๐/๐ ๐ = (๐โ๐)๐^๐ (๐๐". " ๐ฅ๐จ๐ " " (๐โ๐)" + " ๐^๐/(๐ โ๐)) Now, ๐๐ฆ/๐๐ฅ = ๐๐ข/๐๐ฅ + ๐๐ฃ/๐๐ฅ = ๐^(๐^๐โ ๐) ((๐^๐โ ๐)/๐+๐๐ ๐ฅ๐จ๐ โก๐ ) + (๐โ๐)๐^๐ (๐^๐/(๐ โ๐)+๐๐ .๐ฅ๐จ๐ โก(๐ โ๐) )