To find the second derivative of the function \( y = (2x + 1)^3 \), we follow these steps:

### 1. First derivative:
Given \( y = (2x + 1)^3 \), we'll differentiate it with respect to \( x \).

Using the chain rule:

\[
\frac{dy}{dx} = 3(2x + 1)^2 \cdot \frac{d}{dx}(2x + 1)
\]

The derivative of \( 2x + 1 \) with respect to \( x \) is 2, so:

\[
\frac{dy}{dx} = 3(2x + 1)^2 \cdot 2 = 6(2x + 1)^2
\]

### 2. Second derivative:
Now, we'll differentiate the first derivative \( \frac{dy}{dx} = 6(2x + 1)^2 \) with respect to \( x \).

Again, using the chain rule:

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

The derivative of \( 2x + 1 \) with respect to \( x \) is still 2, so:

\[
\frac{d^2y}{dx^2} = 6 \cdot 2(2x + 1) \cdot 2 = 24(2x + 1)
\]

### Final Answer:
\[
\frac{d^2y}{dx^2} = 24(2x + 1)
\]

This is the second derivative of the function \( y = (2x + 1)^3 \).

Would you like more details or have any questions?

Here are some related questions you might find interesting:
1. How do you apply the chain rule in calculus?
2. What is the significance of the second derivative in graph analysis?
3. How do you find higher-order derivatives of a function?
4. How would the second derivative change if the function were \( y = (3x + 2)^4 \)?
5. What are the applications of derivatives in physics?

**Tip:** The second derivative of a function can give insight into the concavity of its graph, indicating whether the function is concave up or concave down.

Discover how to find the second derivative of the function y = (2x + 1)^3 with detailed steps and explanations. Learn about differentiation, the chain rule, and second-order derivatives. Suitable for college-level calculus students.

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