Differentiate the following functions:
`7x^(6) + 8x^(5) - 3x^(4) + 11x^(2) + 6x+7`

Answer

To differentiate the function \( f(x) = 7x^6 + 8x^5 - 3x^4 + 11x^2 + 6x + 7 \), we will apply the power rule of differentiation, which states that if \( f(x) = x^m \), then \( \frac{d}{dx}(x^m) = mx^{m-1} \). Additionally, the derivative of a constant is zero.

### Step-by-Step Solution:

1. **Identify the function**:
\[ f(x) = 7x^6 + 8x^5 - 3x^4 + 11x^2 + 6x + 7 \]

2. **Differentiate each term**:
- For \( 7x^6 \):
\[ \frac{d}{dx}(7x^6) = 7 \cdot 6x^{6-1} = 42x^5 \]
- For \( 8x^5 \):
\[ \frac{d}{dx}(8x^5) = 8 \cdot 5x^{5-1} = 40x^4 \]
- For \( -3x^4 \):
\[ \frac{d}{dx}(-3x^4) = -3 \cdot 4x^{4-1} = -12x^3 \]
- For \( 11x^2 \):
\[ \frac{d}{dx}(11x^2) = 11 \cdot 2x^{2-1} = 22x \]
- For \( 6x \):
\[ \frac{d}{dx}(6x) = 6 \cdot 1x^{1-1} = 6 \]
- For the constant \( 7 \):
\[ \frac{d}{dx}(7) = 0 \]

3. **Combine all the derivatives**:
\[ f'(x) = 42x^5 + 40x^4 - 12x^3 + 22x + 6 + 0 \]

4. **Final result**:
\[ f'(x) = 42x^5 + 40x^4 - 12x^3 + 22x + 6 \]