To find the derivative of the function $y = 3x^3 + 4x$, we'll differentiate each term with respect to $x$.

Step 1: Differentiate Each Term

1. Differentiate $3x^3$: $\frac{d}{dx}(3x^3) = 3 \times 3x^{3-1} = 9x^2$

2. Differentiate $4x$: $\frac{d}{dx}(4x) = 4 \times 1x^{1-1} = 4$

Step 2: Combine the Results

The derivative of the function $y = 3x^3 + 4x$ is: $\frac{dy}{dx} = 9x^2 + 4$

So, the derivative of the function is $\boxed{9x^2 + 4}$.

Would you like further details or have any questions?

Here are 8 related questions:

1. What is the second derivative of $y = 3x^3 + 4x$?
2. How does the derivative $9x^2 + 4$ describe the slope of the original function?
3. What are the critical points of the function $y = 3x^3 + 4x$?
4. How does the graph of the function $y = 3x^3 + 4x$ compare to its derivative?
5. What is the derivative at a specific point, say $x = 2$?
6. How would you integrate the derivative to get back to the original function?
7. What does the sign of the derivative indicate about the function's behavior?
8. How does the power rule apply in finding the derivative of polynomial functions?

Tip: When differentiating, remember to apply the power rule $\frac{d}{dx}(x^n) = nx^{n-1}$ for each term separately, then combine them to get the final derivative.