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Misc 23 - If y = ea cos-1 x, show (1 - x2) d2y/dx2 - x dy/dx - Finding second order derivatives- Implicit form

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  1. Chapter 5 Class 12 Continuity and Differentiability
  2. Serial order wise
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Misc 23 If ๐‘ฆ=๐‘’^(ใ€–๐‘Ž ๐‘๐‘œ๐‘ ใ€—^(โˆ’1) ๐‘ฅ) , โ€“ 1 โ‰ค ๐‘ฅ โ‰ค 1, show that (1โˆ’๐‘ฅ^2 ) (๐‘‘^2 ๐‘ฆ)/ใ€–๐‘‘๐‘ฅใ€—^2 โˆ’ ๐‘ฅ ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ โˆ’๐‘Ž2 ๐‘ฆ 0 . ๐‘ฆ=๐‘’^(ใ€–๐‘Ž ๐‘๐‘œ๐‘ ใ€—^(โˆ’1) ๐‘ฅ) Let ใ€–๐‘Ž ๐‘๐‘œ๐‘ ใ€—^(โˆ’1) ๐‘ฅ=๐‘ก ๐‘ฆ=๐‘’^๐‘ก Differentiating ๐‘ค.๐‘Ÿ.๐‘ก.๐‘ฅ. ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ = ๐‘‘(๐‘’^๐‘ก )/๐‘‘๐‘ฅ ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ = ๐‘‘(๐‘’^๐‘ก )/๐‘‘๐‘ฅ ร— ๐‘‘๐‘ก/๐‘‘๐‘ก ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ = ๐‘‘(๐‘’^๐‘ก )/๐‘‘๐‘ก ร— ๐‘‘๐‘ก/๐‘‘๐‘ฅ ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ = ๐‘’^๐‘ก ร— ๐‘‘๐‘ก/๐‘‘๐‘ฅ Putting value of ๐‘ก=ใ€–๐‘Ž ๐‘๐‘œ๐‘ ใ€—^(โˆ’1) ๐‘ฅ ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ = ๐‘’^(ใ€–๐‘Ž ๐‘๐‘œ๐‘ ใ€—^(โˆ’1) ๐‘ฅ" " ) ร— ๐‘‘(ใ€–๐‘Ž ๐‘๐‘œ๐‘ ใ€—^(โˆ’1) ๐‘ฅ)/๐‘‘๐‘ฅ ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ = ๐‘’^(ใ€–๐‘Ž ๐‘๐‘œ๐‘ ใ€—^(โˆ’1) ๐‘ฅ" " ) ร— ๐‘Ž ๐‘‘(ใ€–๐‘๐‘œ๐‘ ใ€—^(โˆ’1) ๐‘ฅ)/๐‘‘๐‘ฅ ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ = ๐‘’^(ใ€–๐‘Ž ๐‘๐‘œ๐‘ ใ€—^(โˆ’1) ๐‘ฅ" " ) ร— ๐‘Ž ((โˆ’1)/โˆš(1 โˆ’ ๐‘ฅ^2 )) ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ = (โˆ’๐‘Ž ๐‘’^(ใ€–๐‘Ž ๐‘๐‘œ๐‘ ใ€—^(โˆ’1) ๐‘ฅ" " ))/โˆš(1 โˆ’ ๐‘ฅ^2 ) Again Differentiating ๐‘ค.๐‘Ÿ.๐‘ก.๐‘ฅ. ๐‘‘/๐‘‘๐‘ฅ (๐‘‘๐‘ฆ/๐‘‘๐‘ฅ) = ๐‘‘/๐‘‘๐‘ฅ ((โˆ’๐‘Ž ๐‘’^(ใ€–๐‘Ž ๐‘๐‘œ๐‘ ใ€—^(โˆ’1) ๐‘ฅ" " ))/โˆš(1 โˆ’ ๐‘ฅ^2 ))

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Davneet Singh
Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He provides courses for Mathematics from Class 9 to 12. You can ask questions here.
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