To differentiate \( \sin^2(4x) \) with respect to \(x\), we will apply both the **chain rule** and the **power rule**.

### Step-by-Step Solution:

Given: \( y = \sin^2(4x) \)

1. Use the chain rule: Rewrite the function as \( y = (\sin(4x))^2 \).
2. Apply the power rule first:
   \[
   \frac{d}{dx} (\sin(4x))^2 = 2 \cdot \sin(4x) \cdot \frac{d}{dx}(\sin(4x))
   \]
3. Differentiate \( \sin(4x) \) with respect to \(x\) using the chain rule again:
   \[
   \frac{d}{dx}(\sin(4x)) = \cos(4x) \cdot \frac{d}{dx}(4x) = 4\cos(4x)
   \]
4. Now substitute this result back into the equation:
   \[
   \frac{d}{dx} \sin^2(4x) = 2 \cdot \sin(4x) \cdot 4 \cos(4x)
   \]
5. Simplify:
   \[
   \frac{d}{dx} \sin^2(4x) = 8 \sin(4x) \cos(4x)
   \]
6. Using the double-angle identity \( \sin(2\theta) = 2\sin(\theta)\cos(\theta) \), we can rewrite the result:
   \[
   \frac{d}{dx} \sin^2(4x) = 4 \sin(8x)
   \]

Thus, the derivative of \( \sin^2(4x) \) with respect to \(x\) is:
\[
\frac{d}{dx} \sin^2(4x) = 4 \sin(8x)
\]

Would you like any further details or clarifications?

### Related Questions:
1. How would you differentiate \( \sin^3(4x) \)?
2. What is the derivative of \( \cos^2(5x) \)?
3. How does the chain rule work in general for trigonometric functions?
4. Can you explain the double-angle identities in more detail?
5. How do you find the derivative of composite functions like \( \tan^2(3x) \)?

### Tip:
When differentiating trigonometric functions raised to powers, always start by breaking them down into simpler parts using the chain rule.

This detailed solution explains how to differentiate sin²(4x) using the chain rule, power rule, and double-angle identity. Learn how to apply these calculus concepts with a step-by-step breakdown, suitable for advanced high school or undergraduate students.

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