The set of points where `f(x)=x/(1+|x|)` is differentiable is

Options

A. `(-oo,-1) , cup (-1, oo)`
B. `(-oo,oo)`
C. `(0, oo)`
D. `(-oo , 0) cup ( 0, oo)`

Answer

To determine the set of points where the function \( f(x) = \frac{x}{1 + |x|} \) is differentiable, we will analyze the function piecewise due to the absolute value.

### Step 1: Define the function piecewise
The function can be expressed as:
\[ f(x) = \begin{cases} \frac{x}{1 + x} & \text{if } x \geq 0 \\ \frac{x}{1 - x} & \text{if } x < 0 \end{cases} \]

### Step 2: Check continuity at \( x = 0 \)
To check if \( f(x) \) is differentiable at \( x = 0 \), we first need to ensure it is continuous at that point. We will find the left-hand limit (LHL), right-hand limit (RHL), and the value of the function at \( x = 0 \).

**Left-hand limit (LHL) as \( x \to 0^- \):**
\[ \text{LHL} = \lim_{x \to 0^-} f(x) = \lim_{x \to 0^-} \frac{x}{1 - x} = \frac{0}{1} = 0 \]

**Right-hand limit (RHL) as \( x \to 0^+ \):**
\[ \text{RHL} = \lim_{x \to 0^+} f(x) = \lim_{x \to 0^+} \frac{x}{1 + x} = \frac{0}{1} = 0 \]

**Value of the function at \( x = 0 \):**
\[ f(0) = \frac{0}{1 + 0} = 0 \]

Since LHL = RHL = \( f(0) = 0 \), we conclude that \( f(x) \) is continuous at \( x = 0 \).

### Step 3: Find the derivative for \( x \neq 0 \)
Next, we differentiate \( f(x) \) for \( x > 0 \) and \( x < 0 \).

**For \( x \geq 0 \):**
\[ f'(x) = \frac{(1 + x)(1) - x(1)}{(1 + x)^2} = \frac{1}{(1 + x)^2} \]

**For \( x < 0 \):**
\[ f'(x) = \frac{(1 - x)(1) - x(-1)}{(1 - x)^2} = \frac{1 + x}{(1 - x)^2} \]

### Step 4: Check differentiability at \( x = 0 \)
We need to check if the left-hand derivative (LHD) and right-hand derivative (RHD) at \( x = 0 \) are equal.

**Right-hand derivative (RHD) at \( x = 0 \):**
\[ \text{RHD} = \lim_{x \to 0^+} f'(x) = \lim_{x \to 0^+} \frac{1}{(1 + x)^2} = 1 \]

**Left-hand derivative (LHD) at \( x = 0 \):**
\[ \text{LHD} = \lim_{x \to 0^-} f'(x) = \lim_{x \to 0^-} \frac{1 + x}{(1 - x)^2} = \frac{1 + 0}{(1 - 0)^2} = 1 \]

Since LHD = RHD = 1, \( f(x) \) is differentiable at \( x = 0 \).

### Conclusion
The function \( f(x) \) is continuous and differentiable everywhere on the real line.

### Final Answer
The set of points where \( f(x) \) is differentiable is \( (-\infty, \infty) \).

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