` sqrt(cot^(-1)sqrt(x))`

Answer

To differentiate the function \( y = \sqrt{\cot^{-1}(\sqrt{x})} \) with respect to \( x \), we will use the chain rule. Here’s a step-by-step solution:

### Step 1: Identify the outer and inner functions
Let:
- \( u = \cot^{-1}(\sqrt{x}) \)
- Therefore, \( y = \sqrt{u} \)

### Step 2: Differentiate the outer function
Using the chain rule, we differentiate \( y \) with respect to \( u \):
\[ \frac{dy}{du} = \frac{1}{2\sqrt{u}} = \frac{1}{2\sqrt{\cot^{-1}(\sqrt{x})}} \]

### Step 3: Differentiate the inner function
Next, we differentiate \( u \) with respect to \( x \):
\[ u = \cot^{-1}(\sqrt{x}) \]
Using the derivative of \( \cot^{-1}(x) \), we have:
\[ \frac{du}{dx} = -\frac{1}{1 + (\sqrt{x})^2} \cdot \frac{d}{dx}(\sqrt{x}) \]
Now, differentiate \( \sqrt{x} \):
\[ \frac{d}{dx}(\sqrt{x}) = \frac{1}{2\sqrt{x}} \]
Thus, substituting this into our derivative of \( u \):
\[ \frac{du}{dx} = -\frac{1}{1 + x} \cdot \frac{1}{2\sqrt{x}} = -\frac{1}{2\sqrt{x}(1 + x)} \]

### Step 4: Apply the chain rule
Now, we can find \( \frac{dy}{dx} \) using the chain rule:
\[ \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} \]
Substituting the derivatives we found:
\[ \frac{dy}{dx} = \frac{1}{2\sqrt{\cot^{-1}(\sqrt{x})}} \cdot \left(-\frac{1}{2\sqrt{x}(1 + x)}\right) \]

### Step 5: Simplify the expression
This simplifies to:
\[ \frac{dy}{dx} = -\frac{1}{4\sqrt{x}\sqrt{\cot^{-1}(\sqrt{x})}(1 + x)} \]

### Final Answer
Thus, the derivative of \( y = \sqrt{\cot^{-1}(\sqrt{x})} \) with respect to \( x \) is:
\[ \frac{dy}{dx} = -\frac{1}{4\sqrt{x}\sqrt{\cot^{-1}(\sqrt{x})}(1 + x)} \]