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The given function is:

\[
y = \left( 4 + \cos^2(x) \right)^8
\]

To find the derivative \( \frac{dy}{dx} \), we will use the chain rule.

1. Let \( u = 4 + \cos^2(x) \), so the equation becomes \( y = u^8 \).
2. The derivative of \( y = u^8 \) with respect to \( u \) is \( \frac{dy}{du} = 8u^7 \).
3. Now, we need to find \( \frac{du}{dx} \). Since \( u = 4 + \cos^2(x) \), we use the chain rule again:
   \[
   \frac{du}{dx} = 2 \cos(x) (-\sin(x)) = -2\cos(x)\sin(x)
   \]
4. Therefore, the full derivative is:
   \[
   \frac{dy}{dx} = 8u^7 \cdot \frac{du}{dx} = 8(4 + \cos^2(x))^7 \cdot (-2\cos(x)\sin(x))
   \]
   Simplifying:
   \[
   \frac{dy}{dx} = -16(4 + \cos^2(x))^7 \cdot \cos(x)\sin(x)
   \]

This is the derivative of the given function.

Would you like a deeper explanation or have any questions?

Here are 5 related questions:
1. How does the chain rule apply to more complex functions?
2. What are common mistakes when using the product and chain rule together?
3. Can you differentiate \( \sin^n(x) \) in a similar fashion?
4. How do you approach higher-order derivatives of trigonometric functions?
5. How can you use trigonometric identities to simplify expressions like this?

**Tip:** Always be mindful of negative signs when differentiating trigonometric functions—small mistakes can lead to large errors!

Explore how to differentiate the function y = (4 + cos²(x))^8 step-by-step using the chain rule and trigonometric differentiation. This solution provides a detailed breakdown suitable for advanced high school or early college calculus students.

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