Find dy/dx by implicit differentiation. sqrt(5x+y) = 6+x^2y^2

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sqrt(5x+y) = 6 + x^2 *y^2

Square both sides:

==> (5x + y ) = 36 + 12x^2 y^2 + x^4 y^4

==> x^4y^4 + 12x^2 y^2 - 5x - y + 36 = 0

==> (xy)^4 + 12(xy)^2 - 5x -y + 36 = 0

Now let us differentiate:

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sqrt(5x+y) = 6 + x^2 *y^2

Square both sides:

==> (5x + y ) = 36 + 12x^2 y^2 + x^4 y^4

==> x^4y^4 + 12x^2 y^2 - 5x - y + 36 = 0

==> (xy)^4 + 12(xy)^2 - 5x -y + 36 = 0

Now let us differentiate:

==> 4(xy)^3 (xy)' + 24(xy)(xy)' - 5 - y' + 0 = 0

==> 4(xy)^3 (y+xy') + 24(xy)(y+xy') -5 - y' = 0

==> 4x^3 y^4 + 4x^4 y^3 y' + 24xy^2 + 24x^ 2y y' - 5 - y' = 0

Now keep terms with y' on one side:

==> 4x^4 y^3 y' + 24x^2 y y' - y' = 5- 24xy^2 - 4x^3 y^4

Now factor y':

==> y'(4x^4 y^3 + 24x^2 y -1) = (5-24xy^2 -4x^3 y^4)

==> y' = (5-24xy^2-4x^3y^4)/(4x^4 y^3 + 24x^2 y - 1)

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