To find the derivative of the function $f(x) = 3x^3 + 4x + 2$, we will apply the rules of differentiation:

1. Power Rule: If $f(x) = ax^n$, then $f'(x) = n \cdot ax^{n-1}$.
2. Constant Rule: The derivative of a constant is zero.
3. Sum Rule: The derivative of a sum is the sum of the derivatives.

Now, let's differentiate each term in $f(x)$:

1. Derivative of $3x^3$ is $9x^2$ (using the power rule).
2. Derivative of $4x$ is $4$ (since $x^1$ becomes $1x^0 = 1$).
3. Derivative of the constant $2$ is $0$.

Combining these results, we get: $f'(x) = 9x^2 + 4$

Would you like more details or have any other questions?

Here are 5 related questions for further exploration:

1. What is the derivative of $f(x) = 5x^4 + 3x^2 + 7$?
2. How does the derivative change if we add a constant to the function?
3. What is the second derivative of $f(x) = 3x^3 + 4x + 2$?
4. How can we use derivatives to find the slope of the tangent line at a specific point on the curve?
5. What is the geometric interpretation of a derivative?

Tip: When differentiating a sum of functions, always apply the differentiation rules to each term separately.