Ex 5.3, 6 - Chapter 5 Class 12 Continuity and Differentiability

Transcript

Ex 5.3, 6 Find 𝑑𝑦﷮𝑑𝑥﷯ in, 𝑥3 + 𝑥2𝑦 + 𝑥𝑦2 + 𝑦3 = 81 𝑥3 + 𝑥2𝑦 + 𝑥𝑦2 + 𝑦3 = 81 Differentiating both sides 𝑤.𝑟.𝑡.𝑥 . 𝑑 𝑥3 + 𝑥2𝑦 + 𝑥𝑦2 + 𝑦3﷯﷮𝑑𝑥﷯ = 𝑑 81﷯﷮𝑑𝑥﷯ 𝑑 𝑥3﷯﷮𝑑𝑥﷯ + 𝑑 𝑥2𝑦﷯﷮𝑑𝑥﷯ + 𝑑(𝑥𝑦2)﷮𝑑𝑥﷯ + 𝑑 𝑦3﷯﷮𝑑𝑥﷯ =0 3 𝑥﷮3−1﷯ + 𝑑(𝑥2𝑦)﷮𝑑𝑥﷯ + 𝑑(𝑥𝑦2)﷮𝑑𝑥﷯ + 𝑑 𝑦3﷯﷮𝑑𝑥﷯ × 𝑑𝑦﷮𝑑𝑦﷯ = 0 3 𝑥﷮2﷯ + 𝑑(𝑥2𝑦)﷮𝑑𝑥﷯ + 𝑑(𝑥𝑦2)﷮𝑑𝑥﷯ + 𝑑 𝑦3﷯﷮𝑑𝑥﷯ × 𝑑𝑦﷮𝑑𝑥﷯ = 0 3 𝑥﷮2﷯+ 𝑑(𝑥2𝑦)﷮𝑑𝑥﷯ + 𝑑(𝑥𝑦2)﷮𝑑𝑥﷯ +3 𝑦﷮3−1﷯ . 𝑑𝑦﷮𝑑𝑥﷯ = 0 3𝑥2 + 𝑑(𝑥2𝑦)﷮𝑑𝑥﷯ + 𝑑(𝑥𝑦2)﷮𝑑𝑥﷯ +3 𝑦﷮2﷯ 𝑑𝑦﷮𝑑𝑥﷯ = 0 3𝑥2 + 𝑑(𝑥2)﷮𝑑𝑥﷯.𝑦+𝑥2 . 𝑑(𝑦)﷮𝑑𝑥﷯﷯+ 𝑑(𝑥)﷮𝑑𝑥﷯.𝑦2+ . 𝑑(𝑦2)﷮𝑑𝑥﷯𝑥 ﷯+ 3y2 𝑑𝑦﷮ 𝑑𝑥﷯ = 0 3𝑥2 + 2𝑥.𝑦+𝑥2 𝑑(𝑦)﷮𝑑𝑥﷯﷯ + 1.𝑦2+𝑥 . 𝑑(𝑦2)﷮𝑑𝑥﷯ ﷯+ 3y2 𝑑𝑦﷮ 𝑑𝑥﷯ = 0 3𝑥2 + 2𝑥.𝑦+𝑥2 𝑑(𝑦)﷮𝑑𝑥﷯﷯ + 𝑦2+𝑥 . 𝑑(𝑦2)﷮𝑑𝑥﷯ × 𝑑𝑦﷮𝑑𝑦﷯﷯+ 3y2 𝑑𝑦﷮ 𝑑𝑥﷯ = 0 3𝑥2 + 2𝑥𝑦+𝑥2 𝑑𝑦﷮𝑑𝑥﷯+𝑦2+𝑥. 𝑑(𝑦2)﷮𝑑𝑦﷯ × 𝑑𝑦﷮𝑑𝑥﷯+3𝑦2 𝑑𝑦﷮ 𝑑𝑥﷯ = 0 3𝑥2 + 2𝑥𝑦+𝑥2 𝑑𝑦﷮𝑑𝑥﷯+𝑦2 + 𝑥. 2 𝑦﷮2−1﷯ 𝑑𝑦﷮𝑑𝑥﷯﷯+3𝑦2 𝑑𝑦﷮𝑑𝑥﷯=0 3𝑥2 + 2𝑥𝑦+𝑦2 + x2 𝑑𝑦﷮𝑑𝑥﷯+𝑥.2𝑦 𝑑𝑦﷮𝑑𝑥﷯+3𝑦2 𝑑𝑦﷮𝑑𝑥﷯=0 (3𝑥2 + 2𝑥𝑦+𝑦2) + 𝑑𝑦﷮𝑑𝑥﷯ 𝑥2+2𝑥𝑦+3𝑦2﷯=0 𝑑𝑦﷮𝑑𝑥﷯ 𝑥2+2𝑥𝑦+3𝑦2﷯=− (3𝑥2 + 2𝑥𝑦+𝑦2) 𝒅𝒚﷮𝒅𝒙﷯= − (𝟑𝒙2 + 𝟐𝒙𝒚 + 𝒚𝟐)﷮ 𝒙𝟐 + 𝟐𝒙𝒚 + 𝟑𝒚𝟐﷯﷯