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# Find dy/dx  in the following: x ^3 +x ^2 y+ x y ^2 + y ^ 3

6. Find dy/dx  in the following:

$x ^3 + x^2 y + xy^2 + y^3 = 81$

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Given function is
$x ^3 + x^2 y + xy^2 + y^3 = 81$
We can rewrite it as
$x^2 y + xy^2 + y^3 = 81 - x^3$
Now, differentiation w.r.t. x is
$\frac{d(x^2 y + xy^2 + y^3)}{dx} = \frac{d(81 - x^3)}{dx}$
$2xy+y^2+\frac{dy}{dx}(x^2+2xy+3y^2) = -3x^2\\ \frac{dy}{dx}=\frac{-(3x^2+2xy+y^2)}{(x^2+2xy+3y^2}$
Therefore, the answer is  $\frac{-(3x^2+2xy+y^2)}{(x^2+2xy+3y^2}$

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