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