Use implicit differentiation to find $d y / d x$. Then find the slope of the curve at the given point.
\[
5 x y-8 x+y=40 ; \quad\left(-4,-\frac{8}{19}\right)
\]
\[
\frac{d y}{d x}=\frac{8-5 y}{5 x+1}
\]
The slope of the curve at $\left(-4,-\frac{8}{19}\right)$ is $\square$.

Answer

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Answer

Final Answer: The slope of the curve at $\left(-4,-\frac{8}{19}\right)$ is $\boxed{-0.531855955678670}$.

Steps

Step 1 ：Use implicit differentiation to find $\frac{d y}{d x}$.

Step 2 ：$\frac{d y}{d x}=\frac{8-5 y}{5 x+1}$

Step 3 ：Substitute the x and y coordinates of the given point into the derivative to find the slope at that point.

Step 4 ：$\frac{d y}{d x} = \frac{8 - 5*(-\frac{8}{19})}{5*(-4) + 1}$

Step 5 ：Simplify the expression to find the slope.

Step 6 ：The slope of the curve at $\left(-4,-\frac{8}{19}\right)$ is $-0.531855955678670$.

Step 7 ：Final Answer: The slope of the curve at $\left(-4,-\frac{8}{19}\right)$ is $\boxed{-0.531855955678670}$.