Differentiate the following functions:
11. `2x^((1)/(2)) + 6x^((1)/(3)) - 2x^((3)/(2))`

Answer

To differentiate the function \( f(x) = 2x^{\frac{1}{2}} + 6x^{\frac{1}{3}} - 2x^{\frac{3}{2}} \), 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**:
\[ f(x) = 2x^{\frac{1}{2}} + 6x^{\frac{1}{3}} - 2x^{\frac{3}{2}} \]

2. **Differentiate each term separately**:
- For the first term \( 2x^{\frac{1}{2}} \):
\[ \frac{d}{dx}(2x^{\frac{1}{2}}) = 2 \cdot \frac{1}{2} x^{\frac{1}{2} - 1} = x^{-\frac{1}{2}} \]
- For the second term \( 6x^{\frac{1}{3}} \):
\[ \frac{d}{dx}(6x^{\frac{1}{3}}) = 6 \cdot \frac{1}{3} x^{\frac{1}{3} - 1} = 2x^{-\frac{2}{3}} \]
- For the third term \( -2x^{\frac{3}{2}} \):
\[ \frac{d}{dx}(-2x^{\frac{3}{2}}) = -2 \cdot \frac{3}{2} x^{\frac{3}{2} - 1} = -3x^{\frac{1}{2}} \]

3. **Combine the derivatives**:
\[ f'(x) = x^{-\frac{1}{2}} + 2x^{-\frac{2}{3}} - 3x^{\frac{1}{2}} \]

4. **Rewrite the expression**:
- The final derivative can be expressed in a more standard form:
\[ f'(x) = \frac{1}{x^{\frac{1}{2}}} + \frac{2}{x^{\frac{2}{3}}} - 3x^{\frac{1}{2}} \]

### Final Answer:
\[ f'(x) = \frac{1}{x^{\frac{1}{2}}} + \frac{2}{x^{\frac{2}{3}}} - 3x^{\frac{1}{2}} \]