Which one of the following functions is differentiable for all real values of x?

Options

A. `(x)/(|x|)`
B. `x|x|`
C. `(1)/(|x|)`
D. `(1)/(x)`

Answer

To determine which function is differentiable for all real values of \( x \), we will analyze the given options step by step.

### Step 1: Analyze the Functions
We need to check the differentiability of each function at all points, particularly at points where the function might not be continuous or differentiable, such as \( x = 0 \).

### Step 2: Check Option A: \( f(x) = \frac{x}{|x|} \)
1. **Definition of the function**:
- For \( x > 0 \): \( f(x) = \frac{x}{x} = 1 \)
- For \( x < 0 \): \( f(x) = \frac{x}{-x} = -1 \)
- For \( x = 0 \): \( f(x) \) is undefined.

2. **Check continuity at \( x = 0 \)**:
- The left-hand limit as \( x \) approaches 0 from the negative side: \( \lim_{x \to 0^-} f(x) = -1 \)
- The right-hand limit as \( x \) approaches 0 from the positive side: \( \lim_{x \to 0^+} f(x) = 1 \)
- Since the left-hand limit and right-hand limit are not equal, \( f(x) \) is not continuous at \( x = 0 \).

3. **Conclusion**: Since \( f(x) \) is not continuous at \( x = 0 \), it is not differentiable at this point.

### Step 3: Check Option B: \( f(x) = |x|^2 \)
1. **Definition of the function**:
- For all \( x \): \( f(x) = x^2 \) (since \( |x|^2 = x^2 \) for all \( x \)).

2. **Check differentiability**:
- The function \( f(x) = x^2 \) is a polynomial function, which is differentiable everywhere on the real line.

3. **Conclusion**: Since \( f(x) = |x|^2 \) is differentiable for all \( x \), it is a candidate for our answer.

### Step 4: Check Options C and D
1. **Option C**: \( f(x) = \frac{1}{x} \)
- This function is not defined at \( x = 0 \) and thus is not differentiable there.

2. **Option D**: \( f(x) = \sqrt{x} \)
- This function is not defined for \( x < 0 \) and thus is not differentiable for all real values of \( x \).

### Final Conclusion
After analyzing all the options, we conclude that the only function that is differentiable for all real values of \( x \) is:

**Answer**: Option B: \( f(x) = |x|^2 \)

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