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We need to differentiate \( y = 8 \cos(16x^4) \).

### Steps:

1. **Differentiate the outer function**:
   \[
   y = 8 \cos(16x^4)
   \]
   The derivative of \( \cos(u) \) is \( -\sin(u) \), and the constant 8 comes along:
   \[
   \frac{d}{dx}[8 \cos(16x^4)] = 8 \times (-\sin(16x^4)) \times \frac{d}{dx}[16x^4]
   \]

2. **Differentiate the inner function**:
   \[
   u = 16x^4
   \]
   The derivative of \( 16x^4 \) is:
   \[
   \frac{d}{dx}[16x^4] = 64x^3
   \]

3. **Apply the chain rule**:
   Multiply the derivatives:
   \[
   y' = -8 \sin(16x^4) \times 64x^3
   \]
   Simplifying:
   \[
   y' = -512x^3 \sin(16x^4)
   \]

Thus, the correct derivative is:
\[
y' = -512x^3 \sin(16x^4)
\]

The correct answer from the options provided is: 
\[
y' = -512x^3 \sin(16x^4)
\]

Would you like further explanation on any part? Here are some additional questions related to this problem:

1. How is the chain rule applied in differentiation?
2. What are the general rules for differentiating trigonometric functions?
3. How can you verify the result using implicit differentiation?
4. What would the derivative be if the function had a different trigonometric term, such as \( \tan(16x^4) \)?
5. How would you differentiate the inverse trigonometric functions, like \( \arcsin(x) \)?

**Tip**: Always simplify the derivative step-by-step and check for mistakes in applying the chain rule.

This problem demonstrates how to differentiate y = 8 cos(16x^4) using the chain rule. Learn how to apply trigonometric differentiation rules step-by-step, including the power rule and the chain rule, to obtain the derivative.

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