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Question
(x2 + y2)2 = xy
Solution
Given that: (x2 + y2)2 = xy
⇒ x4 + y4 + 2x2y2 = xy
Differentiating both sides w.r.t. x
`"d"/"dx"(x^4) + "d"/"dx"(y^4) + 2*"d"/"dx"(x^2y^2) = "d"/"dx"(xy)`
⇒ `4x^3 + 4y^3 * "dy"/"dx" + 2[x^2*2y*"dy"/"dx" + y^2*2x] = x"dy"/"dx" + y*1`
⇒ `4x^3 + 4y^3 * "dy"/"dx" + 4x^2y * "dy"/"dx" + 4xy^2 = x "dy"/"dx" + y`
⇒ `4y^3 "dy"/"dx" + 4x^2y "dy"/"dx" - x "dy"/"dx" = y - 4x^3 - 4xy^2`
⇒ `(4y^3 + 4x^2y - x)"dy"/"dx" = y - 4x^3 - 4xy^2`
⇒ `"dy"/"dx" = (y - 4x^3 - 4xy^2)/(4y^3 + 4x^2y - x)`
Hence, `"dy"/"dx" = (y - 4x^3 - 4xy^2)/(4x^2y + 4x^2y - x)`.
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