Differentiate implicitly to find $\frac{d y}{d x}$ . Then, find the slope of the curve at the given point.
$x^{2} - 3 y^{2} = - 4; (\sqrt{5}, \sqrt{3})$
$\frac{d y}{d x} =$

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Answer

So, the slope of the curve at the given point $(\sqrt{5}, \sqrt{3})$ is $\frac{\sqrt{15}}{9}$ .

Steps

Step 1 ：Given the equation $x^{2} - 3 y^{2} = - 4$ , we can differentiate both sides with respect to $x$ .

Step 2 ：Applying the power rule, we get $2 x - 3 \cdot 2 y \cdot \frac{d y}{d x} = 0$ .

Step 3 ：Rearranging the equation, we get $\frac{d y}{d x} = \frac{2 x}{6 y}$ .

Step 4 ：Simplifying the equation, we get $\frac{d y}{d x} = \frac{x}{3 y}$ .

Step 5 ：Substitute the given point $(\sqrt{5}, \sqrt{3})$ into the equation, we get $\frac{d y}{d x} = \frac{\sqrt{5}}{3 \sqrt{3}}$ .

Step 6 ：Simplify the equation, we get $\frac{d y}{d x} = \frac{\sqrt{15}}{9}$ .

Step 7 ：So, the slope of the curve at the given point $(\sqrt{5}, \sqrt{3})$ is $\frac{\sqrt{15}}{9}$ .