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

Answer

To differentiate the function \( y = \cot^{-1}(\sqrt{x}) \) with respect to \( x \), we will follow these steps:

### Step 1: Set up the equation
Let \( y = \cot^{-1}(\sqrt{x}) \).

### Step 2: Differentiate using the chain rule
We know that the derivative of \( \cot^{-1}(u) \) with respect to \( u \) is given by:
\[ \frac{dy}{du} = -\frac{1}{1 + u^2} \]
where \( u = \sqrt{x} \).

### Step 3: Differentiate \( u = \sqrt{x} \)
Next, we need to find \( \frac{du}{dx} \):
\[ u = \sqrt{x} = x^{1/2} \]
Using the power rule, we differentiate:
\[ \frac{du}{dx} = \frac{1}{2} x^{-1/2} = \frac{1}{2\sqrt{x}} \]

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

### Step 5: Simplify the expression
Since \( (\sqrt{x})^2 = x \), we can simplify:
\[ \frac{dy}{dx} = -\frac{1}{1 + x} \cdot \frac{1}{2\sqrt{x}} = -\frac{1}{2\sqrt{x}(1 + x)} \]

### Final Result
Thus, the derivative of \( y = \cot^{-1}(\sqrt{x}) \) with respect to \( x \) is:
\[ \frac{dy}{dx} = -\frac{1}{2\sqrt{x}(1 + x)} \]

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