To find the derivative of the function \( f(x) = x|x| \) at \( x = 1 \), we need to break the function into cases due to the absolute value.

### Step 1: Rewrite the function using piecewise notation
The function \( f(x) = x|x| \) can be written as:

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

This piecewise definition makes it easier to handle the derivative.

### Step 2: Differentiate each piece
Now, we differentiate each piece of the function separately:

- For \( x \geq 0 \), \( f(x) = x^2 \), so:
  \[
  f'(x) = 2x
  \]

- For \( x < 0 \), \( f(x) = -x^2 \), so:
  \[
  f'(x) = -2x
  \]

### Step 3: Evaluate the derivative at \( x = 1 \)
Since \( 1 \geq 0 \), we use the derivative from the \( x \geq 0 \) case, which is \( f'(x) = 2x \). Therefore, at \( x = 1 \):
\[
f'(1) = 2(1) = 2
\]

Thus, the derivative of \( f(x) = x|x| \) at \( x = 1 \) is \( f'(1) = 2 \).

---

Would you like more details on the differentiation process, or do you have any questions?

Here are 5 related questions you might find interesting:
1. What is the derivative of \( f(x) = x|x| \) at \( x = -1 \)?
2. How would you compute the second derivative of \( f(x) = x|x| \)?
3. Can we use the chain rule to differentiate absolute value functions?
4. What is the general formula for the derivative of \( |x| \)?
5. How do you differentiate functions with absolute values at points of non-differentiability?

**Tip**: Always check the continuity and differentiability of piecewise functions at the boundary points, such as where the absolute value changes definition!

Explore the step-by-step solution for finding the derivative of f(x) = x|x| at x = 1. Learn about piecewise functions, handling absolute values, and applying differentiation rules. This problem is suitable for advanced high school or early college students.

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