Differentiate the following w.r.t. x :
`cot^(-1)(sqrt(1+x^(2))-x)`.

Answer

To differentiate the function \( y = \cot^{-1}(\sqrt{1+x^2} - x) \) with respect to \( x \), we will follow these steps:

### Step 1: Identify the function to differentiate
We have:
\[ y = \cot^{-1}(\sqrt{1+x^2} - x) \]

### Step 2: Differentiate using the chain rule
Using the chain rule, the derivative of \( \cot^{-1}(u) \) is given by:
\[ \frac{dy}{dx} = -\frac{1}{1 + u^2} \cdot \frac{du}{dx} \]
where \( u = \sqrt{1+x^2} - x \).

### Step 3: Find \( \frac{du}{dx} \)
First, we need to differentiate \( u \):
\[ u = \sqrt{1+x^2} - x \]
Differentiating \( u \):
\[ \frac{du}{dx} = \frac{d}{dx}(\sqrt{1+x^2}) - \frac{d}{dx}(x) \]
Using the derivative of \( \sqrt{1+x^2} \):
\[ \frac{d}{dx}(\sqrt{1+x^2}) = \frac{1}{2\sqrt{1+x^2}} \cdot 2x = \frac{x}{\sqrt{1+x^2}} \]
Thus,
\[ \frac{du}{dx} = \frac{x}{\sqrt{1+x^2}} - 1 \]

### Step 4: Substitute \( u \) and \( \frac{du}{dx} \) into the derivative formula
Now substituting \( u \) and \( \frac{du}{dx} \) into the derivative formula:
\[ \frac{dy}{dx} = -\frac{1}{1 + (\sqrt{1+x^2} - x)^2} \cdot \left( \frac{x}{\sqrt{1+x^2}} - 1 \right) \]

### Step 5: Simplify \( 1 + u^2 \)
Next, we simplify \( 1 + u^2 \):
\[ u^2 = (\sqrt{1+x^2} - x)^2 = 1 + x^2 - 2x\sqrt{1+x^2} \]
Thus,
\[ 1 + u^2 = 1 + (1 + x^2 - 2x\sqrt{1+x^2}) = 2 + x^2 - 2x\sqrt{1+x^2} \]

### Step 6: Final expression for \( \frac{dy}{dx} \)
Putting it all together, we have:
\[ \frac{dy}{dx} = -\frac{1}{2 + x^2 - 2x\sqrt{1+x^2}} \cdot \left( \frac{x}{\sqrt{1+x^2}} - 1 \right) \]

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