The domain of the derivative of the function `f(x)={{:(tan^(-1)x ,if|x|lt=1), (1/2(|x|-1),if|x|>1`):}

Options

A. (a)`R - {0}`
B. (b)`R - {1}`
C. (c)`R - {-1}`
D. (d)`R - {-1, 1}`

Answer

To find the domain of the derivative of the function

\[ f(x) = \begin{cases} \tan^{-1}(x) & \text{if } |x| \leq 1 \\ \frac{1}{2}(|x| - 1) & \text{if } |x| > 1 \end{cases} \]

we will analyze the function piece by piece and find its derivative.

### Step 1: Identify the pieces of the function

The function is defined in two parts:
1. For \(|x| \leq 1\), \(f(x) = \tan^{-1}(x)\)
2. For \(|x| > 1\), \(f(x) = \frac{1}{2}(|x| - 1)\)

### Step 2: Find the derivative for each piece

**For \(|x| \leq 1\):**
- The derivative of \(f(x) = \tan^{-1}(x)\) is:
\[ f'(x) = \frac{1}{1+x^2} \]
This is valid for \(-1 \leq x \leq 1\).

**For \(|x| > 1\):**
- The function can be simplified as:
\[ f(x) = \begin{cases} \frac{1}{2}(x - 1) & \text{if } x > 1 \\ \frac{1}{2}(-x - 1) & \text{if } x < -1 \end{cases} \]
- The derivative for \(x > 1\) is:
\[ f'(x) = \frac{1}{2} \]
- The derivative for \(x < -1\) is:
\[ f'(x) = -\frac{1}{2} \]

### Step 3: Determine the points of interest

The function \(f(x)\) is continuous at \(x = 1\) and \(x = -1\). We need to check the derivatives at these points to determine the domain of \(f'(x)\).

### Step 4: Check continuity and differentiability at the transition points

- At \(x = 1\):
- \(f'(1) = \frac{1}{1+1^2} = \frac{1}{2}\) from the left.
- \(f'(1) = \frac{1}{2}\) from the right.

- At \(x = -1\):
- \(f'(-1) = \frac{1}{1+(-1)^2} = \frac{1}{2}\) from the left.
- \(f'(-1) = -\frac{1}{2}\) from the right.

Since the left-hand derivative and the right-hand derivative at \(x = -1\) do not match, \(f'(x)\) does not exist at \(x = -1\).

### Step 5: Conclusion on the domain of \(f'(x)\)

The derivative \(f'(x)\) exists for all \(x\) except at \(x = -1\). Therefore, the domain of the derivative \(f'(x)\) is:

\[ \text{Domain of } f'(x) = \mathbb{R} \setminus \{-1\} \]

### Final Answer

The domain of the derivative of the function \(f(x)\) is \(\mathbb{R} \setminus \{-1\}\).

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