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.