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We are asked to find the derivative of the function:

\[
y = \left( \frac{x^2 + 2}{x^2 - 2} \right)^4
\]

This function involves the chain rule and the quotient rule. Let’s break it down step by step.

### Step 1: Apply the Chain Rule
We have a function raised to the 4th power, so we first use the chain rule. The derivative of \( y \) with respect to \( x \) starts with differentiating the outer function:

\[
y' = 4 \left( \frac{x^2 + 2}{x^2 - 2} \right)^3 \cdot \frac{d}{dx} \left( \frac{x^2 + 2}{x^2 - 2} \right)
\]

Now, we need to find the derivative of the inner function \( \frac{x^2 + 2}{x^2 - 2} \).

### Step 2: Apply the Quotient Rule
To differentiate \( \frac{x^2 + 2}{x^2 - 2} \), we use the quotient rule:

\[
\frac{d}{dx} \left( \frac{x^2 + 2}{x^2 - 2} \right) = \frac{(x^2 - 2) \cdot \frac{d}{dx}(x^2 + 2) - (x^2 + 2) \cdot \frac{d}{dx}(x^2 - 2)}{(x^2 - 2)^2}
\]

Let’s compute the derivatives in the numerator:

- \( \frac{d}{dx}(x^2 + 2) = 2x \)
- \( \frac{d}{dx}(x^2 - 2) = 2x \)

Substitute these into the quotient rule formula:

\[
\frac{d}{dx} \left( \frac{x^2 + 2}{x^2 - 2} \right) = \frac{(x^2 - 2)(2x) - (x^2 + 2)(2x)}{(x^2 - 2)^2}
\]

### Step 3: Simplify the Derivative
Simplify the numerator:

\[
(2x)(x^2 - 2) - (2x)(x^2 + 2) = 2x(x^2 - 2 - x^2 - 2) = 2x(-4) = -8x
\]

Thus, the derivative of the inner function is:

\[
\frac{d}{dx} \left( \frac{x^2 + 2}{x^2 - 2} \right) = \frac{-8x}{(x^2 - 2)^2}
\]

### Step 4: Combine Results
Now substitute this back into the chain rule expression:

\[
y' = 4 \left( \frac{x^2 + 2}{x^2 - 2} \right)^3 \cdot \frac{-8x}{(x^2 - 2)^2}
\]

Simplify the expression:

\[
y' = -32x \cdot \frac{\left( x^2 + 2 \right)^3}{\left( x^2 - 2 \right)^5}
\]

So, the derivative of the function is:

\[
y' = -32x \cdot \frac{\left( x^2 + 2 \right)^3}{\left( x^2 - 2 \right)^5}
\]

Let me know if you need more details or have any questions!

Here are some related questions to explore further:
1. How would you apply the product rule to functions with multiple terms?
2. Can you explain the difference between the chain rule and the power rule?
3. How does the quotient rule work for more complex expressions?
4. What happens when you differentiate a logarithmic function?
5. How would implicit differentiation apply if the variables were mixed?

**Tip**: Always simplify the function as much as possible before applying the rules of differentiation.

Find the derivative of the function y = ((x^2 + 2)/(x^2 - 2))^4.

Learn how to find the derivative of the function ((x^2 + 2)/(x^2 - 2))^4 using the chain rule and quotient rule. This detailed solution is perfect for students studying calculus and higher-level mathematics. Suitable for Grades 11-12.

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