Check Full Chapter Explained - Continuity and Differentiability - https://you.tube/Chapter-5-Class-12-Continuity

Slide21.JPG

Slide22.JPG
Slide23.JPG

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

Transcript

Example 41 If y = ใ€–๐‘ ๐‘–๐‘›ใ€—^(โˆ’1) ๐‘ฅ, show that (1 โ€“ ๐‘ฅ2) ๐‘‘2๐‘ฆ/๐‘‘๐‘ฅ2 โˆ’ ๐‘ฅ ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ = 0 . We have ๐‘ฆ = ใ€–๐‘ ๐‘–๐‘›ใ€—^(โˆ’1) ๐‘ฅ Differentiating ๐‘ค.๐‘Ÿ.๐‘ก.๐‘ฅ ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ = ๐‘‘(ใ€–๐‘ ๐‘–๐‘›ใ€—^(โˆ’1) ๐‘ฅ)/๐‘‘๐‘ฅ ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ = 1/โˆš(ใ€–1 โˆ’ ๐‘ฅใ€—^2 ) ๐’…๐’š/๐’…๐’™ = (ใ€–๐Ÿโˆ’๐’™ใ€—^๐Ÿ )^((โˆ’๐Ÿ)/( ๐Ÿ)) ("As " ๐‘‘(ใ€–๐‘ ๐‘–๐‘›ใ€—^(โˆ’1) ๐‘ฅ)/๐‘‘๐‘ฅ " = " 1/โˆš(ใ€–1 โˆ’ ๐‘ฅใ€—^2 )) Again Differentiating ๐‘ค.๐‘Ÿ.๐‘ก.๐‘ฅ ๐‘‘/๐‘‘๐‘ฅ (๐‘‘๐‘ฆ/๐‘‘๐‘ฅ) = (๐‘‘(ใ€–1 โˆ’ ๐‘ฅใ€—^2 )^((โˆ’1)/( 2)))/๐‘‘๐‘ฅ (๐‘‘^2 ๐‘ฆ)/ใ€–๐‘‘๐‘ฅใ€—^2 = (โˆ’1)/( 2) (ใ€–1โˆ’๐‘ฅใ€—^2 )^((โˆ’1)/( 2) โˆ’1) . ๐‘‘(ใ€–1 โˆ’ ๐‘ฅใ€—^2 )/๐‘‘๐‘ฅ (๐‘‘^2 ๐‘ฆ)/ใ€–๐‘‘๐‘ฅใ€—^2 = (โˆ’1)/( 2) (ใ€–1โˆ’๐‘ฅใ€—^2 )^((โˆ’3)/2 ). (0โˆ’2๐‘ฅ) (๐‘‘^2 ๐‘ฆ)/ใ€–๐‘‘๐‘ฅใ€—^2 = (โˆ’1)/( 2) (ใ€–1โˆ’๐‘ฅใ€—^2 )^((โˆ’3)/2 ). (โˆ’2๐‘ฅ) (๐’…^๐Ÿ ๐’š)/ใ€–๐’…๐’™ใ€—^๐Ÿ = ๐’™(ใ€–๐Ÿโˆ’๐’™ใ€—^๐Ÿ )^((โˆ’๐Ÿ‘)/๐Ÿ ) Now, we need to prove (ใ€–1โˆ’๐‘ฅใ€—^2 ) (๐‘‘^2 ๐‘ฆ)/ใ€–๐‘‘๐‘ฅใ€—^2 โˆ’ ๐‘ฅ . ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ = 0 Solving LHS (ใ€–1โˆ’๐‘ฅใ€—^2 ) (๐‘‘^2 ๐‘ฆ)/ใ€–๐‘‘๐‘ฅใ€—^2 โˆ’ ๐‘ฅ . ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ = (ใ€–1โˆ’๐‘ฅใ€—^2 ) . (๐‘ฅใ€– (ใ€–1โˆ’๐‘ฅใ€—^2 )ใ€—^((โˆ’3)/2 ) ) โˆ’ ๐‘ฅ (ใ€–1โˆ’๐‘ฅใ€—^2 )^((โˆ’1)/( 2)) = ๐‘ฅใ€– (ใ€–1โˆ’๐‘ฅใ€—^2 )ใ€—^(1+ ((โˆ’3)/2) )โˆ’๐‘ฅ (ใ€–1โˆ’๐‘ฅใ€—^2 )^((โˆ’1)/( 2)) = ๐‘ฅใ€– (ใ€–1โˆ’๐‘ฅใ€—^2 )ใ€—^((โˆ’1)/( 2))โˆ’๐‘ฅ (ใ€–1โˆ’๐‘ฅใ€—^2 )^((โˆ’1)/( 2)) = 0 = RHS Hence proved

About the Author

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.