What is the derivative of `f(x)=x|x|`?

Options

A. `|x|+x`
B. `2x`
C. `2|x|`
D. `-2|x|`

Explanation

To find the derivative of the function \( f(x) = x |x| \), we start by understanding the absolute value function. The absolute value function \( |x| \) is defined as follows:

\[ |x| = \begin{cases} x & \text{if } x \geq 0 \\ -x & \text{if } x < 0 \end{cases} \]

Using this definition, we can rewrite the function \( f(x) \) in piecewise form:

\[ f(x) = \begin{cases} x^2 & \text{if } x \geq 0 \\ -x^2 & \text{if } x < 0 \end{cases} \]

Now, we will differentiate \( f(x) \) for both cases.

### Step 1: Differentiate for \( x \geq 0 \)

For \( x \geq 0 \):
\[ f(x) = x^2 \]
The derivative is:
\[ f'(x) = \frac{d}{dx}(x^2) = 2x \]

### Step 2: Differentiate for \( x < 0 \)

For \( x < 0 \):
\[ f(x) = -x^2 \]
The derivative is:
\[ f'(x) = \frac{d}{dx}(-x^2) = -2x \]

### Step 3: Combine the results

Now we can combine the results from both cases into a piecewise function for the derivative:

\[ f'(x) = \begin{cases} 2x & \text{if } x \geq 0 \\ -2x & \text{if } x < 0 \end{cases} \]

### Step 4: Express in terms of absolute value

Notice that \( -2x \) can also be expressed as \( 2|x| \) when \( x < 0 \). Thus, we can write:

\[ f'(x) = 2|x| \]

### Final Result

The derivative of the function \( f(x) = x|x| \) is:

\[ f'(x) = 2|x| \]