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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

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Transcript

Ex 5.7, 8 Find the second order derivatives of the function 〖𝑡𝑎𝑛〗^(−1) 𝑥 Let y = 〖𝑡𝑎𝑛〗^(−1) 𝑥 Differentiating 𝑤.𝑟.𝑡.𝑥 . 𝑑𝑦/𝑑𝑥 = (𝑑(〖𝑡𝑎𝑛〗^(−1) 𝑥))/𝑑𝑥 𝑑𝑦/𝑑𝑥 = 1/(1 + 𝑥^2 ) Again Differentiating 𝑤.𝑟.𝑡.𝑥 𝑑/𝑑𝑥 (𝑑𝑦/𝑑𝑥) = 𝑑/𝑑𝑥 (1/(1 + 𝑥^2 )) (𝑑^2 𝑦)/(𝑑𝑥^2 ) = 𝑑/𝑑𝑥 (1/(1 + 𝑥^2 )) Using Quotient Rule As, (((𝑢)′)/𝑣) = (𝑢’𝑣 − 𝑣’𝑢)/𝑣^2 where u = 1 & v = 1 + x2 (𝑑^2 𝑦)/(𝑑𝑥^2 ) = ((𝑑(1))/𝑑𝑥 (1+𝑥^2 ) − (𝑑 (1 +𝑥^2 ))/𝑑𝑥 . 1 )/(1+𝑥^2 )^2 (𝑑^2 𝑦)/(𝑑𝑥^2 ) = (0 . (1+𝑥^2 ) − ((𝑑(1))/𝑑𝑥 + (𝑑〖(𝑥〗^2))/𝑑𝑥). 1 )/(1+𝑥^2 )^2 (𝑑^2 𝑦)/(𝑑𝑥^2 ) = (0 − ( 0 + 2𝑥 ) 1)/(1+𝑥^2 )^2 (𝒅^𝟐 𝒚)/(𝒅𝒙^𝟐 ) = (−𝟐𝒙)/(𝟏+𝒙^𝟐 )^𝟐

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Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.