We are given the equation:

$2x^2 - 3y^2 = 2$

Step 1: Differentiate both sides implicitly

Differentiate both terms with respect to $x$:

$\frac{d}{dx} (2x^2) - \frac{d}{dx} (3y^2) = \frac{d}{dx} (2)$

Using the power rule:

$4x - 6y \frac{dy}{dx} = 0$

Step 2: Solve for $\frac{dy}{dx}$

Rearrange the equation:

$-6y \frac{dy}{dx} = -4x$

$\frac{dy}{dx} = \frac{4x}{6y} = \frac{2x}{3y}$

Step 3: Find values at $x = 5$

We substitute $x = 5$ into the original equation to find $y$:

$2(5^2) - 3y^2 = 2$

$50 - 3y^2 = 2$

$3y^2 = 48$

$y^2 = 16$

$y = \pm 4$

Step 4: Compute $\frac{dy}{dx}$ for both values of $y$

For $y = 4$:

$\frac{dy}{dx} = \frac{2(5)}{3(4)} = \frac{10}{12} = \frac{5}{6}$

For $y = -4$:

$\frac{dy}{dx} = \frac{2(5)}{3(-4)} = \frac{10}{-12} = -\frac{5}{6}$

Final Answer:

The two values of $\frac{dy}{dx}$ when $x = 5$ are:

$\frac{5}{6}, \quad -\frac{5}{6}$

Would you like a detailed explanation of any step?

Here are five related questions to explore further:

1. How do you apply implicit differentiation to more complex equations?
2. What is the geometric interpretation of having two values for $\frac{dy}{dx}$?
3. How does implicit differentiation differ from explicit differentiation?
4. Can you use a different method to find $\frac{dy}{dx}$ for this problem?
5. What happens if we solve for $y$ explicitly before differentiating?

Tip: Always check for both positive and negative values of $y$ when solving for derivatives implicitly!