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!