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.