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Question
Find `(dy)/(dx)`, if x3 + x2y + xy2 + y3 = 81
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Solution
x3 + x3y + xy2 + y3 = 81
Differentiating both sides w.r.t x. we get
`= d/dx (x^3) + d/dx (x^2y) + d/dx (xy^2) + d/dx (y^3) = d/dx (81)`
`= 3x^2 + x^2 xx(d(y))/dx + yxx(d(x^2))/dx + xxx(d(y^2))/dx + y^2xx(d(x))/dx+3y^2xxdy/dx = 0`
`= 3x^2 + x^2dy/dx + yxx2x + x xx2y dy/dx + y^2xx1 + 3y^2 dy/dx`
`= dy/dx (x^2 + 2xy + 3y^2) = -3x^2-2xy-y^2`
`therefore dy/dx= -(3x^2 + 2xy + y^2)/(x^2+2xy+3y^2)`
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