(i) ` 4x^(3) + 3.2^(x) + 6.root8(x^(-4)) + 5 cot x` (ii) ` x/3 -3/x +sqrtx - 1/sqrtx + x^(2) -2^(x) +6x^(-2//3) -2/3 x^(6)`

Answer

To find the derivatives of the given functions, we will apply the rules of differentiation step by step.

### Part (i): Differentiate \( 4x^3 + 3 \cdot 2^x + 6 \cdot \sqrt[8]{x^{-4}} + 5 \cot x \)

1. **Differentiate \( 4x^3 \)**:
\[ \frac{d}{dx}(4x^3) = 4 \cdot 3x^{3-1} = 12x^2 \]

2. **Differentiate \( 3 \cdot 2^x \)**:
\[ \frac{d}{dx}(3 \cdot 2^x) = 3 \cdot 2^x \cdot \log(2) \]

3. **Differentiate \( 6 \cdot \sqrt[8]{x^{-4}} \)**:
Rewrite \( \sqrt[8]{x^{-4}} \) as \( 6 \cdot x^{-4/8} = 6 \cdot x^{-1/2} \).
\[ \frac{d}{dx}(6 \cdot x^{-1/2}) = 6 \cdot \left(-\frac{1}{2}x^{-3/2}\right) = -3x^{-3/2} \]

4. **Differentiate \( 5 \cot x \)**:
\[ \frac{d}{dx}(5 \cot x) = 5 \cdot (-\csc^2 x) = -5 \csc^2 x \]

5. **Combine all the derivatives**:
\[ f'(x) = 12x^2 + 3 \cdot 2^x \cdot \log(2) - 3x^{-3/2} - 5 \csc^2 x \]

### Final Result for Part (i):
\[ f'(x) = 12x^2 + 3 \cdot 2^x \cdot \log(2) - 3x^{-3/2} - 5 \csc^2 x \]

---

### Part (ii): Differentiate \( \frac{x}{3} - \frac{3}{x} + \sqrt{x} - \frac{1}{\sqrt{x}} + x^2 - 2^x + 6x^{-2/3} - \frac{2}{3}x^6 \)

1. **Differentiate \( \frac{x}{3} \)**:
\[ \frac{d}{dx}\left(\frac{x}{3}\right) = \frac{1}{3} \]

2. **Differentiate \( -\frac{3}{x} \)**:
Rewrite as \( -3x^{-1} \).
\[ \frac{d}{dx}(-3x^{-1}) = 3x^{-2} \]

3. **Differentiate \( \sqrt{x} \)**:
Rewrite as \( x^{1/2} \).
\[ \frac{d}{dx}(x^{1/2}) = \frac{1}{2}x^{-1/2} \]

4. **Differentiate \( -\frac{1}{\sqrt{x}} \)**:
Rewrite as \( -x^{-1/2} \).
\[ \frac{d}{dx}(-x^{-1/2}) = \frac{1}{2}x^{-3/2} \]

5. **Differentiate \( x^2 \)**:
\[ \frac{d}{dx}(x^2) = 2x \]

6. **Differentiate \( -2^x \)**:
\[ \frac{d}{dx}(-2^x) = -2^x \cdot \log(2) \]

7. **Differentiate \( 6x^{-2/3} \)**:
\[ \frac{d}{dx}(6x^{-2/3}) = 6 \cdot \left(-\frac{2}{3}x^{-5/3}\right) = -4x^{-5/3} \]

8. **Differentiate \( -\frac{2}{3}x^6 \)**:
\[ \frac{d}{dx}\left(-\frac{2}{3}x^6\right) = -\frac{2}{3} \cdot 6x^{5} = -4x^{5} \]

9. **Combine all the derivatives**:
\[ g'(x) = \frac{1}{3} + 3x^{-2} + \frac{1}{2}x^{-1/2} + \frac{1}{2}x^{-3/2} + 2x - 2^x \cdot \log(2) - 4x^{-5/3} - 4x^5 \]

### Final Result for Part (ii):
\[ g'(x) = \frac{1}{3} + 3x^{-2} + \frac{1}{2}x^{-1/2} + \frac{1}{2}x^{-3/2} + 2x - 2^x \cdot \log(2) - 4x^{-5/3} - 4x^5 \]

---