Differentiate implicitly to find $d y / d x$. Then find the slope of the curve at the given point.
\[
x^{2} y^{2}=100, \quad(-5,2)
\]
\[
\frac{d y}{d x}=
\]

Answer

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Answer

So, the slope of the curve at the point $(-5,2)$ is $\boxed{\frac{2}{5}}$.

Steps

Step 1 ：Given the equation $x^{2} y^{2}=100$, we can differentiate both sides with respect to $x$.

Step 2 ：Applying the product rule to $x^{2} y^{2}$, we get $2x y^{2} + x^{2} 2y \frac{dy}{dx} = 0$.

Step 3 ：Rearranging the equation, we get $\frac{dy}{dx} = -\frac{2xy^{2}}{2yx^{2}} = -\frac{y}{x}$.

Step 4 ：Substituting the given point $(-5,2)$ into the equation, we get $\frac{dy}{dx} = -\frac{2}{-5} = \frac{2}{5}$.

Step 5 ：So, the slope of the curve at the point $(-5,2)$ is $\boxed{\frac{2}{5}}$.