Find the derivative of the function by using the rules of differentiation.

*f*(*x*) = 

3

*x*3

 + 

4

*x*

 + 2

*f* *'*(*x*) = 


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.

Find the derivative of the function by using the rules of differentiation.

f(x) = 3x^3 + 4x + 2

Learn how to find the derivative of the function f(x) = 3x^3 + 4x + 2 by applying the power, constant, and sum rules of differentiation. This step-by-step guide is suitable for high school students in Grades 10-12.

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