The function given is:

\[
y = 3 \sin^3(4x^2 + 1)
\]

This can be understood as \( y = 3 \left( \sin(4x^2 + 1) \right)^3 \).

### Derivative:
To differentiate this function \( y \) with respect to \( x \), we apply the chain rule.

1. **Outer function**: \( y = 3 u^3 \), where \( u = \sin(4x^2 + 1) \).
   - The derivative of \( 3u^3 \) with respect to \( u \) is \( 9u^2 \).

2. **Middle function**: \( u = \sin(4x^2 + 1) \).
   - The derivative of \( \sin(v) \) with respect to \( v \) is \( \cos(v) \), where \( v = 4x^2 + 1 \).

3. **Inner function**: \( v = 4x^2 + 1 \).
   - The derivative of \( 4x^2 + 1 \) with respect to \( x \) is \( 8x \).

Now, putting it all together using the chain rule:

\[
\frac{dy}{dx} = 9 \left( \sin(4x^2 + 1) \right)^2 \cdot \cos(4x^2 + 1) \cdot 8x
\]

Simplifying the expression:

\[
\frac{dy}{dx} = 72x \left( \sin(4x^2 + 1) \right)^2 \cdot \cos(4x^2 + 1)
\]

This is the derivative of the function.

### Do you want details on any step? Here are some related questions to explore:

1. How do you apply the chain rule to trigonometric functions?
2. What are the steps to differentiate functions with powers of trigonometric terms?
3. How does the product rule differ from the chain rule in differentiation?
4. Can we find the second derivative of the function? How?
5. How would you graph this function, and what key points should you consider?

**Tip:** When differentiating composite functions, always start from the outermost function and work inward using the chain rule.

This problem involves finding the derivative of y = 3 sin³(4x² + 1) using the chain rule and trigonometric differentiation. The detailed solution walks through each step, making it ideal for students studying calculus or advanced trigonometry.

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