The derivative of $x^{2}$ is $2x$, the derivative of $y^{2}$ is $2y \frac{dy}{dx}$, and the derivative of $\sin(xy^{2})$ is $\cos(xy^{2})(y^{2}+2xy\frac{dy}{dx})$. So, we get:
$$2x + 2y \frac{dy}{dx} + \cos(xy^{2})(y^{2}+2xy\frac{dy}{dx}) = 0$$
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