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

Answer

To differentiate the function \( y = (\cot^{-1} x)^2 \) with respect to \( x \), we will use the chain rule. Here are the steps:

### Step 1: Identify the outer and inner functions
Let \( u = \cot^{-1} x \). Then, we can express \( y \) as:
\[ y = u^2 \]

### Step 2: Differentiate the outer function
Using the power rule, we differentiate \( y \) with respect to \( u \):
\[ \frac{dy}{du} = 2u \]

### Step 3: Differentiate the inner function
Next, we need to differentiate \( u = \cot^{-1} x \) with respect to \( x \). The derivative of \( \cot^{-1} x \) is given by:
\[ \frac{du}{dx} = -\frac{1}{1 + x^2} \]

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

### Step 5: Substitute back for \( u \)
Now, substitute \( u = \cot^{-1} x \) back into the equation:
\[ \frac{dy}{dx} = 2(\cot^{-1} x) \cdot \left(-\frac{1}{1 + x^2}\right) \]
This simplifies to:
\[ \frac{dy}{dx} = -\frac{2 \cot^{-1} x}{1 + x^2} \]

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

---