To find the second derivative of \((2x + 1)^3\), let's proceed step by step:

### Step 1: First derivative
We start by differentiating \((2x + 1)^3\) using the chain rule.

\[
f(x) = (2x + 1)^3
\]

The chain rule tells us that we first differentiate the outer function, which is \( u^3 \), and then multiply by the derivative of the inner function, \( u = 2x + 1 \).

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

The derivative of \(2x + 1\) is just 2, so:

\[
f'(x) = 3(2x + 1)^2 \cdot 2 = 6(2x + 1)^2
\]

### Step 2: Second derivative
Now, we differentiate \(6(2x + 1)^2\) to find the second derivative.

Using the chain rule again:

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

Since the derivative of \(2x + 1\) is 2:

\[
f''(x) = 6 \cdot 2(2x + 1) \cdot 2 = 24(2x + 1)
\]

### Final Answer:
\[
f''(x) = 24(2x + 1)
\]

Would you like more details on any part of this process or have any other questions?

Here are some related questions:
1. How would the result change if the function was \((2x - 1)^3\)?
2. What is the third derivative of \((2x + 1)^3\)?
3. How does the chain rule work for composite functions?
4. How can you apply this process to \((3x + 2)^4\)?
5. What is the general formula for the derivative of \((ax + b)^n\)?

**Tip:** Always remember to apply the chain rule when differentiating composite functions!

This page provides a step-by-step guide to finding the second derivative of the function (2x + 1)³ using the chain rule. The solution breaks down the process of differentiating composite functions, suitable for students in Grades 11-12.

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