`cot^(-1)((sqrt(1-x^(2)))/(x))`

Answer

To differentiate the function \( y = \cot^{-1}\left(\frac{\sqrt{1 - x^2}}{x}\right) \), we can follow these steps:

### Step 1: Rewrite the function
Let:
\[ y = \cot^{-1}\left(\frac{\sqrt{1 - x^2}}{x}\right) \]

### Step 2: Use the substitution
We can use the trigonometric identity by letting \( x = \sin \theta \). Then, we have:
\[ \sqrt{1 - x^2} = \sqrt{1 - \sin^2 \theta} = \cos \theta \]
Thus, we can rewrite \( y \) as:
\[ y = \cot^{-1}\left(\frac{\cos \theta}{\sin \theta}\right) = \cot^{-1}(\cot \theta) \]

### Step 3: Simplify the expression
Since \( \cot^{-1}(\cot \theta) = \theta \) (for \( \theta \) in the appropriate range), we can express \( y \) as:
\[ y = \theta \]

### Step 4: Relate \( \theta \) back to \( x \)
Since \( x = \sin \theta \), we can express \( \theta \) as:
\[ \theta = \sin^{-1}(x) \]

### Step 5: Differentiate \( y \) with respect to \( x \)
Now we can differentiate \( y \):
\[ \frac{dy}{dx} = \frac{d}{dx}(\sin^{-1}(x)) \]
Using the derivative of the inverse sine function:
\[ \frac{dy}{dx} = \frac{1}{\sqrt{1 - x^2}} \]

### Final Answer
Thus, the derivative of the given function is:
\[ \frac{dy}{dx} = \frac{1}{\sqrt{1 - x^2}} \]

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