web analytics

Ex 5.7, 15 - If y = 500 e7x + 600 e-7x, show d2y/dx2 = 49y - Finding second order derivatives- Implicit form

Slide43.JPG
Slide44.JPG

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

Transcript

Ex 5.7, 15 If ๐‘ฆ= ใ€–500๐‘’ใ€—^7๐‘ฅ+ ใ€–600๐‘’ใ€—^(โˆ’7๐‘ฅ), show that ๐‘‘2๐‘ฆ/๐‘‘๐‘ฅ2 = 49๐‘ฆ ๐‘ฆ= ใ€–500๐‘’ใ€—^7๐‘ฅ+ ใ€–600๐‘’ใ€—^(โˆ’7๐‘ฅ) Differentiating ๐‘ค.๐‘Ÿ.๐‘ก.๐‘ฅ ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ = (๐‘‘(ใ€–500๐‘’ใ€—^7๐‘ฅ "+" ใ€–600๐‘’ใ€—^(โˆ’7๐‘ฅ) " " ))/๐‘‘๐‘ฅ ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ = (๐‘‘ (ใ€–500๐‘’ใ€—^7๐‘ฅ))/๐‘‘๐‘ฅ + (๐‘‘ (ใ€–600๐‘’ใ€—^(โˆ’7๐‘ฅ)))/๐‘‘๐‘ฅ ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ = 500 (๐‘‘ (๐‘’^7๐‘ฅ))/๐‘‘๐‘ฅ + 600 (๐‘‘ (๐‘’^(โˆ’7๐‘ฅ)))/๐‘‘๐‘ฅ ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ = 500 . ๐‘’^7๐‘ฅ. (๐‘‘ (7๐‘ฅ))/๐‘‘๐‘ฅ + 600 . ๐‘’^(โˆ’7๐‘ฅ) . (๐‘‘ (โˆ’7๐‘ฅ))/๐‘‘๐‘ฅ ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ = 500 . ๐‘’^7๐‘ฅ. 7 + 600 . ๐‘’^(โˆ’7๐‘ฅ) . (โˆ’7) ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ = 500 . 7 . ๐‘’^7๐‘ฅ โˆ’ 600 . 7 . ๐‘’^(โˆ’7๐‘ฅ) Again Differentiating ๐‘ค.๐‘Ÿ.๐‘ก.๐‘ฅ ๐‘‘/๐‘‘๐‘ฅ (๐‘‘๐‘ฆ/๐‘‘๐‘ฅ) = ๐‘‘(500 . 7 . ๐‘’^7๐‘ฅ " โˆ’" ใ€– 600 . 7 . ๐‘’ใ€—^(โˆ’7๐‘ฅ) )" " /๐‘‘๐‘ฅ (๐‘‘^2 ๐‘ฆ)/(๐‘‘๐‘ฅ^2 ) = ๐‘‘(500 ร— 7 . ๐‘’^7๐‘ฅ )/๐‘‘๐‘ฅ + ๐‘‘(ใ€–600 ร—7๐‘’ใ€—^(โˆ’7๐‘ฅ) )" " /๐‘‘๐‘ฅ (๐‘‘^2 ๐‘ฆ)/(๐‘‘๐‘ฅ^2 ) = 500 ร— 7 ๐‘‘(๐‘’^7๐‘ฅ )/๐‘‘๐‘ฅ โˆ’ 600 ร— 7 ๐‘‘(๐‘’^(โˆ’7๐‘ฅ) )/๐‘‘๐‘ฅ = 500 ร— 7 ร— ๐‘’^7๐‘ฅ ๐‘‘(7๐‘ฅ)/๐‘‘๐‘ฅ โˆ’ 600 ร— 7 ร— ๐‘’^(โˆ’7๐‘ฅ) ๐‘‘(โˆ’7๐‘ฅ)/๐‘‘๐‘ฅ = 500 ร— 7 ร— ๐‘’^7๐‘ฅ. 7 โˆ’ 600 ร— 7 ร— ๐‘’^(โˆ’7๐‘ฅ) (โˆ’7) = 500 ร— 7 ร— 7๐‘’^7๐‘ฅ โˆ’ 600 ร— 7 ร— 7 ร— ๐‘’^(โˆ’7๐‘ฅ) = 7 ร— 7 (500ใ€– ๐‘’ใ€—^7๐‘ฅ+600ใ€– ๐‘’ใ€—^(โˆ’7๐‘ฅ) ) = 49 (500ใ€– ๐‘’ใ€—^7๐‘ฅ+600ใ€– ๐‘’ใ€—^(โˆ’7๐‘ฅ) ) = 49 ๐‘ฆ 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 provides courses for Mathematics from Class 9 to 12. You can ask questions here.
Jail