6. Find dy/dx  in the following: 

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

Answers (1)

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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