Find the derivatives of the following (1-3) functions :
`f(x)=x^(3)+4x^(2)+3x+2`

Answer

To find the derivative of the function \( f(x) = x^3 + 4x^2 + 3x + 2 \), we will apply the power rule of differentiation. The power rule states that if \( y = x^n \), then the derivative \( \frac{dy}{dx} = nx^{n-1} \).

### Step-by-Step Solution:

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

2. **Differentiate each term separately**:
- For the first term \( x^3 \):
\[ \frac{d}{dx}(x^3) = 3x^{3-1} = 3x^2 \]
- For the second term \( 4x^2 \):
\[ \frac{d}{dx}(4x^2) = 4 \cdot 2x^{2-1} = 8x \]
- For the third term \( 3x \):
\[ \frac{d}{dx}(3x) = 3 \cdot 1x^{1-1} = 3 \]
- For the constant term \( 2 \):
\[ \frac{d}{dx}(2) = 0 \]

3. **Combine the derivatives**:
Now, we will add the derivatives of all the terms together:
\[ f'(x) = 3x^2 + 8x + 3 + 0 \]

4. **Final result**:
Therefore, the derivative of the function \( f(x) \) is:
\[ f'(x) = 3x^2 + 8x + 3 \]