`(2cotx)/(sqrt(x))`

Answer

To differentiate the function \( f(x) = \frac{2 \cot x}{\sqrt{x}} \), we will use the quotient rule. The quotient rule states that if you have a function in the form \( \frac{U}{V} \), then its derivative is given by:

\[ \frac{d}{dx} \left( \frac{U}{V} \right) = \frac{V \frac{dU}{dx} - U \frac{dV}{dx}}{V^2} \]

### Step-by-Step Solution

1. **Identify U and V**:
- Let \( U = 2 \cot x \)
- Let \( V = \sqrt{x} \)

2. **Differentiate U and V**:
- The derivative of \( U \) is:
\[ \frac{dU}{dx} = 2 \cdot (-\csc^2 x) = -2 \csc^2 x \]
- The derivative of \( V \) is:
\[ \frac{dV}{dx} = \frac{1}{2\sqrt{x}} \]

3. **Apply the Quotient Rule**:
- Using the quotient rule:
\[ \frac{d}{dx} \left( \frac{2 \cot x}{\sqrt{x}} \right) = \frac{\sqrt{x} \cdot (-2 \csc^2 x) - 2 \cot x \cdot \frac{1}{2\sqrt{x}}}{(\sqrt{x})^2} \]

4. **Simplify the Expression**:
- The denominator \( (\sqrt{x})^2 = x \).
- The numerator becomes:
\[ -2 \sqrt{x} \csc^2 x - \frac{2 \cot x}{2\sqrt{x}} = -2 \sqrt{x} \csc^2 x - \frac{\cot x}{\sqrt{x}} \]
- Thus, we have:
\[ \frac{-2 \sqrt{x} \csc^2 x - \frac{\cot x}{\sqrt{x}}}{x} \]

5. **Combine Terms**:
- Rewrite the numerator:
\[ \frac{-2 \sqrt{x} \csc^2 x - \cot x/\sqrt{x}}{x} = \frac{-2 \sqrt{x} \csc^2 x - \cot x/\sqrt{x}}{x} \]
- This can be expressed as:
\[ \frac{-2 \sqrt{x} \csc^2 x - \frac{\cot x}{\sqrt{x}}}{x} = \frac{-2 \sqrt{x} \csc^2 x - \frac{\cot x}{\sqrt{x}}}{x} \]

6. **Final Result**:
- The final derivative is:
\[ \frac{-2 \sqrt{x} \csc^2 x - \frac{\cot x}{\sqrt{x}}}{x} \]