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

*f*(*x*) = 

3

*x*3

 + 

4

*x*

 + 2

*f* *'*(*x*) = 

9x2+4
 
 

The function you have is:

\[
f(x) = 3x^3 + 4x + 2
\]

We will differentiate this function using the rules of differentiation, particularly the power rule and the constant rule.

### Step-by-step differentiation:

1. **Differentiate \(3x^3\)**:
   Using the power rule \(\frac{d}{dx}(x^n) = nx^{n-1}\),
   \[
   \frac{d}{dx}(3x^3) = 3 \cdot 3x^{3-1} = 9x^2
   \]

2. **Differentiate \(4x\)**:
   The derivative of \(4x\) (linear term) is just the constant:
   \[
   \frac{d}{dx}(4x) = 4
   \]

3. **Differentiate the constant 2**:
   The derivative of a constant is zero:
   \[
   \frac{d}{dx}(2) = 0
   \]

Now, combine all the terms:

\[
f'(x) = 9x^2 + 4
\]

This matches your result: \(f'(x) = 9x^2 + 4\).

Would you like further details or have any questions?

### Related questions:
1. How would the derivative change if the exponent in the first term were different?
2. What is the derivative of \(f(x) = 5x^4 + 2x^2 + 1\)?
3. Can you find the second derivative of \(f(x)\)?
4. How would you apply the product rule to find the derivative of a product of two functions?
5. How does the chain rule apply when differentiating composite functions?

### Tip:
When differentiating a constant multiplied by a function, the constant can be factored out and then you differentiate the function directly.

Find the derivative of the function by using the rules of differentiation. f(x) = 3x^3 + 4x + 2. f'(x) = 9x^2 + 4

Learn how to find the derivative of the function f(x) = 3x^3 + 4x + 2 using the power rule and other differentiation rules. This solution provides a step-by-step breakdown, ideal for students in Grades 10-12 studying calculus.

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