To find the derivative of the function \( y = (x^2 + 1)^3 \), we will use the **chain rule**.

### Steps:
1. Let \( u = x^2 + 1 \), so that the function becomes \( y = u^3 \).
2. First, differentiate \( y = u^3 \) with respect to \( u \):
   \[
   \frac{dy}{du} = 3u^2
   \]
3. Now, differentiate \( u = x^2 + 1 \) with respect to \( x \):
   \[
   \frac{du}{dx} = 2x
   \]
4. Use the chain rule to combine these derivatives:
   \[
   \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} = 3u^2 \cdot 2x
   \]
5. Substitute \( u = x^2 + 1 \) back into the equation:
   \[
   \frac{dy}{dx} = 3(x^2 + 1)^2 \cdot 2x = 6x(x^2 + 1)^2
   \]

So, the derivative of \( y = (x^2 + 1)^3 \) is:

\[
\frac{dy}{dx} = 6x(x^2 + 1)^2
\]

Would you like a more detailed explanation, or do you have any questions?

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Here are five related questions for further exploration:

1. What is the second derivative of \( y = (x^2 + 1)^3 \)?
2. How do you apply the product rule in differentiation?
3. How does the chain rule apply to more complex functions?
4. Can you find the derivative of \( y = (x^2 + 2)^4 \)?
5. What happens when differentiating a function like \( (x^2 + 1)^n \)?

**Tip:** Always check if the chain rule is needed when differentiating composite functions!

Learn how to differentiate y = (x^2 + 1)^3 using the chain rule. This detailed step-by-step solution explains how to apply the chain rule and power rule to find the derivative. Suitable for calculus students in Grades 11-12.

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