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

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﷮𝑑 𝑥﷮2﷯﷯ = 𝑑﷮𝑑𝑥﷯ 1﷮1+ 𝑥﷮2﷯﷯﷯ 𝑑﷮2﷯𝑦﷮𝑑 𝑥﷮2﷯﷯ = 𝑑(1)﷮𝑑𝑥﷯ 1+ 𝑥﷮2﷯﷯ − 𝑑 1 + 𝑥﷮2﷯﷯﷮𝑑𝑥﷯ . 1 ﷮ 1+ 𝑥﷮2﷯﷯﷮2﷯﷯ 𝑑﷮2﷯𝑦﷮𝑑 𝑥﷮2﷯﷯ = 0 . 1+ 𝑥﷮2﷯﷯ − 𝑑(1)﷮𝑑𝑥﷯ + 𝑑 (𝑥﷮2﷯)﷮𝑑𝑥﷯﷯. 1 ﷮ 1+ 𝑥﷮2﷯﷯﷮2﷯﷯ 𝑑﷮2﷯𝑦﷮𝑑 𝑥﷮2﷯﷯ = 0 − 0 + 2𝑥 ﷯ 1﷮ 1+ 𝑥﷮2﷯﷯﷮2﷯﷯ 𝒅﷮𝟐﷯𝒚﷮𝒅 𝒙﷮𝟐﷯﷯ = −𝟐𝒙﷮ 𝟏+ 𝒙﷮𝟐﷯﷯﷮𝟐﷯﷯