Given the equation:

\[
y^2 - x^2 = 2
\]

We need to find \(\frac{d}{dx}(y)\), or \(\frac{dy}{dx}\), using implicit differentiation. Let's proceed step by step:

### Step 1: Differentiate both sides with respect to \(x\).
Differentiate the left-hand side term by term.

\[
\frac{d}{dx}(y^2 - x^2) = \frac{d}{dx}(2)
\]

Using the chain rule:
- For \(y^2\), treat \(y\) as a function of \(x\) (since \(y = y(x)\)):

\[
\frac{d}{dx}(y^2) = 2y \cdot \frac{dy}{dx}
\]

- For \(x^2\):

\[
\frac{d}{dx}(x^2) = 2x
\]

On the right-hand side, the derivative of the constant \(2\) is 0:

\[
\frac{d}{dx}(2) = 0
\]

### Step 2: Set up the equation.

After differentiating, we get:

\[
2y \cdot \frac{dy}{dx} - 2x = 0
\]

### Step 3: Solve for \(\frac{dy}{dx}\).

First, isolate the term with \(\frac{dy}{dx}\):

\[
2y \cdot \frac{dy}{dx} = 2x
\]

Now, divide both sides by \(2y\):

\[
\frac{dy}{dx} = \frac{x}{y}
\]

Thus, the derivative \(\frac{dy}{dx}\) is:

\[
\frac{dy}{dx} = \frac{x}{y}
\]

---

Would you like more details or have any questions?

Here are 5 related questions for deeper understanding:
1. How would you find \(\frac{d^2y}{dx^2}\) (the second derivative)?
2. What happens if you substitute specific values for \(x\) and \(y\)?
3. How would you interpret the result \(\frac{dy}{dx} = \frac{x}{y}\) graphically?
4. Can you use implicit differentiation if the equation involves higher powers of \(y\)?
5. What is the physical interpretation of \(\frac{dy}{dx} = \frac{x}{y}\) in real-world scenarios?

**Tip:** Implicit differentiation is useful when it’s difficult or impossible to solve for one variable explicitly before differentiating.

Explore how to use implicit differentiation to find the derivative of y with respect to x for the equation y^2 - x^2 = 2. This step-by-step guide covers key differentiation rules like the chain rule and is ideal for high school students in Grades 10-12.

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