The equation y^2e^{xy} = 9e^{-3}x^2 defines y | vedclass

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(D) Given the equation: $y^2e^{xy} = 9e^{-3}x^2$. Differentiating both sides with respect to $x$ using the product rule and chain rule: $\frac{d}{dx}(

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$15$

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(D) Given the equation: $y^2e^{xy} = 9e^{-3}x^2$. Differentiating both sides with respect to $x$ using the product rule and chain rule: $\frac{d}{dx}(y^2)e^{xy} + y^2\frac{d}{dx}(e^{xy}) = 9e^{-3}\frac{d}{dx}(x^2)$ $2y\frac{dy}{dx}e^{xy} + y^2e^{xy}\left(y + x\frac{dy}{dx}ight) = 18e^{-3}x$ Substitute $x = -1$ and $y = 3$: $2(3)\frac{dy}{dx}e^{-3} + (3)^2e^{-3}\left(3 + (-1)\frac{dy}{dx}ight) = 18e^{-3}(-1)$ Divide the entire equation by $e^{-3}$ (since $e^{-3} 
eq 0$): $6\frac{dy}{dx} + 9(3 - \frac{dy}{dx}) = -18$ $6\frac{dy}{dx} + 27 - 9\frac{dy}{dx} = -18$ $-3\frac{dy}{dx} = -18 - 27$ $-3\frac{dy}{dx} = -45$ $\frac{dy}{dx} = 15$.

The equation $y^2e^{xy} = 9e^{-3}x^2$ defines $y$ as a differentiable function of $x$. The value of $\frac{dy}{dx}$ for $x = -1$ and $y = 3$ is

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