The derivative of f(x)=`3x^2` +2x + 4 w.r.t. x is

Options

A. `3(6x+2)^2`
B. `6x+ 4`
C. `3(2x + 3)`
D. `6x + 2`

Answer

To find the derivative of the function \( f(x) = 3x^2 + 2x + 4 \) with respect to \( x \), we will apply the rules of differentiation step by step.

### Step-by-Step Solution:

1. **Identify the function**:
\[ f(x) = 3x^2 + 2x + 4 \]

2. **Differentiate each term**:
- For the first term \( 3x^2 \):
- Use the power rule: \( \frac{d}{dx}(x^n) = n x^{n-1} \).
- Here, \( n = 2 \), so:
\[ \frac{d}{dx}(3x^2) = 3 \cdot 2x^{2-1} = 6x \]

- For the second term \( 2x \):
- Again, apply the power rule where \( n = 1 \):
\[ \frac{d}{dx}(2x) = 2 \cdot 1x^{1-1} = 2 \cdot 1 = 2 \]

- For the third term \( 4 \):
- The derivative of a constant is 0:
\[ \frac{d}{dx}(4) = 0 \]

3. **Combine the derivatives**:
- Now, add the derivatives of each term together:
\[ f'(x) = 6x + 2 + 0 \]
- Thus, we simplify to:
\[ f'(x) = 6x + 2 \]

### Final Answer:
The derivative of \( f(x) = 3x^2 + 2x + 4 \) with respect to \( x \) is:
\[ f'(x) = 6x + 2 \]