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Question
Find `(dy)/(dx)`, if x3 + x3y + xy2 + y3 = 81
Solution
x3 + x3y + xy2 + y3 = 81
Differentiating both sides w.r.t x. we get
`3x^2 + x^2(dy)/(dx) + yd/(dx)(x^2) + xd/(dx)(y^2) + y^2d/(dx)(x) + 3y^2(dy)/(dx)` = 0
∴ `3x^2 + x^2(dy)/(dx) + y xx 2x + x xx 2y(dy)/(dx) + y^2 xx 1 + 3y^2(dy)/(dx)` = 0
∴ `3x^2 + x^2(dy)/(dx) + 2xy + 2xy(dy)/(dx) + y^2 + 3y^2(dy)/(dx)` = 0
∴ `(x^2 + 2xy + 3y^2)(dy)/(dx)` = –3x2 – 2xy – y2
∴ `(dy)/(dx) = (-(3x^2 + 2xy + y^2))/(x^2 + 2xy + 3y^2)`.
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