The derivative of the function f(x) =x|x| at x= 0 is

Options

A. 1
B. 0
C. 2
D. `-2`

Answer

To find the derivative of the function \( f(x) = x |x| \) at \( x = 0 \), we will first express the function in a piecewise manner, then calculate the derivative, and finally evaluate it at \( x = 0 \).

### Step-by-Step Solution:

1. **Understanding the Function**:
The function \( f(x) = x |x| \) can be expressed in terms of piecewise functions based on the definition of absolute value:
\[ f(x) = \begin{cases} x^2 & \text{if } x \geq 0 \\ -x^2 & \text{if } x < 0 \end{cases} \]

2. **Finding the Derivative**:
We will find the derivative of \( f(x) \) for both cases:
- For \( x \geq 0 \):
\[ f(x) = x^2 \implies f'(x) = 2x \]
- For \( x < 0 \):
\[ f(x) = -x^2 \implies f'(x) = -2x \]

3. **Evaluating the Derivative at \( x = 0 \)**:
Now, we need to evaluate the derivative at \( x = 0 \). Since the derivative is defined piecewise, we will check the limit from both sides:
- From the right (as \( x \) approaches 0 from positive values):
\[ f'(0^+) = 2(0) = 0 \]
- From the left (as \( x \) approaches 0 from negative values):
\[ f'(0^-) = -2(0) = 0 \]

4. **Conclusion**:
Since both one-sided derivatives at \( x = 0 \) are equal, we conclude that:
\[ f'(0) = 0 \]

### Final Answer:
The derivative of the function \( f(x) = x |x| \) at \( x = 0 \) is \( 0 \).

---