If the derivative $\frac{d x}{d y}$ (instead of $\frac{d y}{d x}$ ) exists at a point and $\frac{d x}{d y}=0$, then the tangent line at that point is vertical. Calculate $\frac{d x}{d y}$ for the equation $y^{4}+8=5 x^{2}+5 y^{2}$. (Use symbolic notation and fractions where needed. ) \[ \frac{d x}{d y}= \] Find the number of points on the graph where the tangent line is vertical. (Give your answer as a whole number.) number of points:

Step 1 ：Differentiate the given equation \(y^{4}+8=5 x^{2}+5 y^{2}\) with respect to \(y\) to find \(\frac{d x}{d y}\).

Step 2 ：By doing so, we get \(\frac{d x}{d y}=\frac{-4y^3 + 10y}{10x}\).

Step 3 ：Set \(\frac{d x}{d y}=0\) to find the points where the tangent line is vertical.

Step 4 ：Solving the equation \(\frac{-4y^3 + 10y}{10x} = 0\), we find that the solutions are \(y = 0\), \(y = -\sqrt{10}/2\), and \(y = \sqrt{10}/2\).

Step 5 ：Therefore, there are 3 points on the graph where the tangent line is vertical.