The given equation is $x^{2}+x y+y^{2}=4$. The derivative of $x^{2}$ is $2x$, the derivative of $x y$ is $x \frac{dy}{dx} + y$, and the derivative of $y^{2}$ is $2y \frac{dy}{dx}$. The derivative of a constant is zero. So, we have:
$$2x + x \frac{dy}{dx} + y + 2y
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