Find the derivatives of the following functions (1-3) at any point of their domains :
`f(x)=x^(4)+7x^(3)+8x^(2)+3x+2`

Answer

To find the derivative of the function \( f(x) = x^4 + 7x^3 + 8x^2 + 3x + 2 \), we will apply the power rule of differentiation. The power rule states that if \( f(x) = x^n \), then \( f'(x) = n \cdot x^{n-1} \).

### Step-by-Step Solution:

1. **Identify the function**:
\[ f(x) = x^4 + 7x^3 + 8x^2 + 3x + 2 \]

2. **Differentiate each term using the power rule**:
- For \( x^4 \):
\[ \frac{d}{dx}(x^4) = 4x^{4-1} = 4x^3 \]
- For \( 7x^3 \):
\[ \frac{d}{dx}(7x^3) = 7 \cdot 3x^{3-1} = 21x^2 \]
- For \( 8x^2 \):
\[ \frac{d}{dx}(8x^2) = 8 \cdot 2x^{2-1} = 16x \]
- For \( 3x \):
\[ \frac{d}{dx}(3x) = 3 \cdot 1x^{1-1} = 3 \]
- For the constant \( 2 \):
\[ \frac{d}{dx}(2) = 0 \]

3. **Combine all the derivatives**:
\[ f'(x) = 4x^3 + 21x^2 + 16x + 3 + 0 \]

4. **Final derivative**:
\[ f'(x) = 4x^3 + 21x^2 + 16x + 3 \]