Differentiate the following functions:
` 3+ 4x - 7x^(2) - sqrt(2) x^(3) + pi x^(4) - (2)/(5) x^(5) + (4)/(3)`

Answer

To differentiate the function \( f(x) = 3 + 4x - 7x^2 - \sqrt{2} x^3 + \pi x^4 - \frac{2}{5} x^5 + \frac{4}{3} \), we will apply the rules of differentiation step by step.

### Step-by-Step Solution:

1. **Identify the function**:
\[ f(x) = 3 + 4x - 7x^2 - \sqrt{2} x^3 + \pi x^4 - \frac{2}{5} x^5 + \frac{4}{3} \]

2. **Differentiate each term separately**:
- The derivative of a constant is \(0\).
- The derivative of \(kx^n\) is \(k \cdot n \cdot x^{n-1}\).

3. **Differentiate each term**:
- \( \frac{d}{dx}(3) = 0 \)
- \( \frac{d}{dx}(4x) = 4 \)
- \( \frac{d}{dx}(-7x^2) = -7 \cdot 2 \cdot x^{2-1} = -14x \)
- \( \frac{d}{dx}(-\sqrt{2} x^3) = -\sqrt{2} \cdot 3 \cdot x^{3-1} = -3\sqrt{2} x^2 \)
- \( \frac{d}{dx}(\pi x^4) = \pi \cdot 4 \cdot x^{4-1} = 4\pi x^3 \)
- \( \frac{d}{dx}\left(-\frac{2}{5} x^5\right) = -\frac{2}{5} \cdot 5 \cdot x^{5-1} = -2x^4 \)
- \( \frac{d}{dx}\left(\frac{4}{3}\right) = 0 \)

4. **Combine the derivatives**:
\[ f'(x) = 0 + 4 - 14x - 3\sqrt{2} x^2 + 4\pi x^3 - 2x^4 + 0 \]
\[ f'(x) = 4 - 14x - 3\sqrt{2} x^2 + 4\pi x^3 - 2x^4 \]

### Final Answer:
\[ f'(x) = 4 - 14x - 3\sqrt{2} x^2 + 4\pi x^3 - 2x^4 \]