Ex 5.7, 8 - Find second order derivatives of tan-1 x - Ex 5.7

Ex 5.7, 8 - Chapter 5 Class 12 Continuity and Differentiability - Part 2

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

Transcript

Ex 5.7, 8 Find the second order derivatives of the function ใ€–๐‘ก๐‘Ž๐‘›ใ€—^(โˆ’1) ๐‘ฅ Let y = ใ€–๐‘ก๐‘Ž๐‘›ใ€—^(โˆ’1) ๐‘ฅ Differentiating ๐‘ค.๐‘Ÿ.๐‘ก.๐‘ฅ . ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ = (๐‘‘(ใ€–๐‘ก๐‘Ž๐‘›ใ€—^(โˆ’1) ๐‘ฅ))/๐‘‘๐‘ฅ ๐‘‘๐‘ฆ/๐‘‘๐‘ฅ = 1/(1 + 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 (1/(1 + ๐‘ฅ^2 )) (๐‘‘^2 ๐‘ฆ)/(๐‘‘๐‘ฅ^2 ) = ๐‘‘/๐‘‘๐‘ฅ (1/(1 + ๐‘ฅ^2 )) Using Quotient Rule As, (((๐‘ข)โ€ฒ)/๐‘ฃ) = (๐‘ขโ€™๐‘ฃ โˆ’ ๐‘ฃโ€™๐‘ข)/๐‘ฃ^2 where u = 1 & v = 1 + x2 (๐‘‘^2 ๐‘ฆ)/(๐‘‘๐‘ฅ^2 ) = ((๐‘‘(1))/๐‘‘๐‘ฅ (1+๐‘ฅ^2 ) โˆ’ (๐‘‘ (1 +๐‘ฅ^2 ))/๐‘‘๐‘ฅ . 1 )/(1+๐‘ฅ^2 )^2 (๐‘‘^2 ๐‘ฆ)/(๐‘‘๐‘ฅ^2 ) = (0 . (1+๐‘ฅ^2 ) โˆ’ ((๐‘‘(1))/๐‘‘๐‘ฅ + (๐‘‘ใ€–(๐‘ฅใ€—^2))/๐‘‘๐‘ฅ). 1 )/(1+๐‘ฅ^2 )^2 (๐‘‘^2 ๐‘ฆ)/(๐‘‘๐‘ฅ^2 ) = (0 โˆ’ ( 0 + 2๐‘ฅ ) 1)/(1+๐‘ฅ^2 )^2 (๐’…^๐Ÿ ๐’š)/(๐’…๐’™^๐Ÿ ) = (โˆ’๐Ÿ๐’™)/(๐Ÿ+๐’™^๐Ÿ )^๐Ÿ

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Davneet Singh
Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 10 years. He provides courses for Maths and Science at Teachoo.