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Misc 14 - if x = root 1+y + y root 1+x = 0, prove dy/dx - Finding derivative of Implicit functions

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  1. Chapter 5 Class 12 Continuity and Differentiability
  2. Serial order wise
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Misc 14 If 𝑥 ﷮1+𝑦﷯+𝑦 ﷮1+𝑥﷯ = 0 , for –1 < 𝑥 < 1, prove that 𝑑𝑦﷮𝑑𝑥﷯ = 1﷮ 1 + 𝑥﷯2﷯ 𝑥 ﷮1+𝑦﷯+𝑦 ﷮1+𝑥﷯ = 0 𝑥 ﷮1+𝑦﷯ = – 𝑦 ﷮1+𝑥﷯ Squaring both sides 𝑥 ﷮1+𝑦﷯ ﷯﷮2﷯ = −𝑦 ﷮1+𝑥﷯﷯﷮2﷯ 𝑥﷮2﷯ ﷮1+𝑦 ﷯ ﷯﷮2﷯ = −𝑦﷯﷮2﷯ ﷮1+𝑥﷯﷯﷮2﷯ 𝑥﷮2﷯ 1+𝑦﷯ = 𝑦﷮2﷯ 1+𝑥﷯ 𝑥﷮2﷯+ 𝑥﷮2﷯𝑦 = 𝑦﷮2﷯ + 𝑦﷮2﷯𝑥 𝑥﷮2﷯ − 𝑦﷮2﷯ = xy2 − x2y 𝑥 −𝑦﷯ 𝑥+𝑦﷯=𝑥𝑦 𝑦 −𝑥﷯ − 𝑦 −𝑥﷯ 𝑥+𝑦﷯=𝑥𝑦 𝑦 −𝑥﷯ − 𝑥+𝑦﷯ = 𝑥𝑦 𝑦 − 𝑥﷯﷮𝑦 − 𝑥﷯ − 𝑥+𝑦﷯ = 𝑥𝑦 −𝑥 −𝑦 = 𝑥𝑦 −𝑥 = 𝑥𝑦+𝑦 −𝑥 = 𝑥+1﷯ 𝑦 −𝑥﷮𝑥 + 1﷯ = 𝑦 𝑦 = −𝑥﷮𝑥 + 1﷯ Differentiating 𝑤.𝑟.𝑡.𝑥. 𝑑𝑦﷮𝑑𝑥﷯ = 𝑑﷮𝑑𝑥﷯ −𝑥﷮𝑥 + 1﷯﷯ 𝑑𝑦﷮𝑑𝑥﷯ = 𝑑 −𝑥﷯﷮𝑑𝑥﷯ 𝑥 + 1﷯ − 𝑑 𝑥 + 1﷯﷮𝑑𝑥﷯ 8𝑙 . −𝑥﷯﷮ 𝑥 + 1﷯﷮2﷯﷯ 𝑑𝑦﷮𝑑𝑥﷯ = −1 𝑥 + 1﷯ − 1 + 0﷯ 𝑥﷮ 𝑥 + 1﷯﷮2﷯﷯ 𝑑𝑦﷮𝑑𝑥﷯ = −𝑥 − 1 − 𝑥﷮ 𝑥 + 1﷯﷮2﷯﷯ 𝑑𝑦﷮𝑑𝑥﷯ = −1﷮ 𝑥 + 1﷯﷮2﷯﷯ Hence, 𝒅𝒚﷮𝒅𝒙﷯ = −𝟏﷮ 𝒙 + 𝟏﷯﷮𝟐﷯﷯

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CA Maninder Singh is a Chartered Accountant for the past 8 years. He provides courses for Practical Accounts, Taxation and Efiling at teachoo.com .
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