Let f(x) = x|x| then f'(0) is equal to

Options

A. 1
B. `-1`
C. 0
D. `+-1`

Answer

To find \( f'(0) \) for the function \( f(x) = x|x| \), we will follow these steps:

### Step 1: Define the function
The function \( f(x) = x|x| \) can be expressed in piecewise form:
- For \( x \geq 0 \): \( f(x) = x^2 \)
- For \( x < 0 \): \( f(x) = -x^2 \)

### Step 2: Check continuity at \( x = 0 \)
To check if \( f(x) \) is continuous at \( x = 0 \), we need to evaluate \( f(0) \) and the limits from both sides.

- \( f(0) = 0|0| = 0 \)
- Limit as \( x \) approaches 0 from the right:
\[ \lim_{x \to 0^+} f(x) = \lim_{x \to 0^+} x^2 = 0 \]
- Limit as \( x \) approaches 0 from the left:
\[ \lim_{x \to 0^-} f(x) = \lim_{x \to 0^-} -x^2 = 0 \]

Since both limits equal \( f(0) \), the function is continuous at \( x = 0 \).

### Step 3: Find the derivative \( f'(x) \)
Now we find the derivative of \( f(x) \) in both intervals:
- For \( x \geq 0 \):
\[ f'(x) = \frac{d}{dx}(x^2) = 2x \]
- For \( x < 0 \):
\[ f'(x) = \frac{d}{dx}(-x^2) = -2x \]

### Step 4: Evaluate \( f'(0) \)
Now we need to find \( f'(0) \). Since the derivative is defined piecewise, we can use either side to evaluate at \( x = 0 \):
- From the right:
\[ f'(0) = 2(0) = 0 \]
- From the left:
\[ f'(0) = -2(0) = 0 \]

Both approaches give the same result.

### Conclusion
Thus, we find that:
\[ f'(0) = 0 \]