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

### Step 1: Find the first derivative \( \frac{dy}{dx} \).
We apply the chain rule to differentiate \( y = (2x + 1)^3 \).

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

Now, differentiate \( 2x + 1 \):

\[
\frac{d}{dx}(2x + 1) = 2
\]

So, the first derivative is:

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

### Step 2: Find the second derivative \( \frac{d^2y}{dx^2} \).
Differentiate the first derivative \( \frac{dy}{dx} = 6(2x + 1)^2 \) with respect to \( x \):

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

Again, \( \frac{d}{dx}(2x + 1) = 2 \), so:

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

### Final Answer:
The second derivative of the function \( y = (2x + 1)^3 \) is:

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

Would you like more details or have any questions?

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Here are some related questions to consider:

1. What is the first derivative of \( y = (3x + 2)^4 \)?
2. How do you apply the chain rule to differentiate composite functions?
3. What is the second derivative of \( y = (5x - 3)^2 \)?
4. Can you find the second derivative of \( y = \sin(3x + 1) \)?
5. What happens to the second derivative as \( x \) increases for the function \( y = (2x + 1)^3 \)?

**Tip:** Remember, the chain rule is essential when differentiating compositions of functions.

Learn how to find the second derivative of the function (2x + 1)^3 step-by-step. This problem involves differentiation using the chain rule for composite functions. Suitable for advanced high school students.

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