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

Answer

To differentiate the function \( y = (\tan^{-1} x)^2 \) with respect to \( x \), we can use the chain rule. Here’s a step-by-step solution:

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

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

### Step 3: Differentiate the inner function
Next, we need to differentiate \( u \) with respect to \( x \):
\[ \frac{du}{dx} = \frac{d}{dx}(\tan^{-1} x) = \frac{1}{1 + x^2} \]

### Step 4: Apply the chain rule
Using the chain rule, we have:
\[ \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} \]
Substituting the derivatives we found:
\[ \frac{dy}{dx} = 2u \cdot \frac{1}{1 + x^2} \]

### Step 5: Substitute back for \( u \)
Now, substitute back \( u = \tan^{-1} x \):
\[ \frac{dy}{dx} = 2(\tan^{-1} x) \cdot \frac{1}{1 + x^2} \]

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

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