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We are asked to find the slope of the tangent line to the curve \( f(x) = x|x| \) at the point \( (a, a|a|) \), considering three cases: \( a > 0 \), \( a < 0 \), and \( a = 0 \).

### 1. Analyzing the function
The function \( f(x) = x|x| \) can be split into two cases:
- When \( x \geq 0 \), \( |x| = x \), so \( f(x) = x \cdot x = x^2 \).
- When \( x < 0 \), \( |x| = -x \), so \( f(x) = x(-x) = -x^2 \).

### 2. Finding the derivative for different cases
To find the slope of the tangent line, we need the derivative \( f'(x) \) for the two cases.

- **Case 1: \( x > 0 \)**  
  When \( x > 0 \), \( f(x) = x^2 \), and its derivative is:
  \[
  f'(x) = 2x
  \]
  So the slope of the tangent line at \( x = a \) when \( a > 0 \) is:
  \[
  f'(a) = 2a
  \]

- **Case 2: \( x < 0 \)**  
  When \( x < 0 \), \( f(x) = -x^2 \), and its derivative is:
  \[
  f'(x) = -2x
  \]
  So the slope of the tangent line at \( x = a \) when \( a < 0 \) is:
  \[
  f'(a) = -2a
  \]

- **Case 3: \( x = 0 \)**  
  When \( x = 0 \), the function \( f(x) = x|x| \) gives \( f(0) = 0 \). To check the slope at this point, we can analyze the derivative from the left and the right:
  \[
  \lim_{h \to 0^+} \frac{f(h) - f(0)}{h} = \lim_{h \to 0^+} \frac{h^2}{h} = 0
  \]
  \[
  \lim_{h \to 0^-} \frac{f(h) - f(0)}{h} = \lim_{h \to 0^-} \frac{-h^2}{h} = 0
  \]
  So, the slope at \( x = 0 \) is \( 0 \).

### 3. Summary of slopes
The slope of the tangent line at the point \( (a, a|a|) \) is:
- For \( a > 0 \), the slope is \( 2a \).
- For \( a < 0 \), the slope is \( -2a \).
- For \( a = 0 \), the slope is \( 0 \).

Let me know if you need further clarifications or details.

Here are some related questions:
1. How do you calculate the derivative of absolute value functions in general?
2. Why do we need to check both left and right limits for slopes at \( a = 0 \)?
3. What is the geometric interpretation of the slopes in each case?
4. Can you explain how the derivative changes for piecewise-defined functions?
5. How can we confirm if a function is differentiable at a certain point?

**Tip:** For piecewise functions involving absolute values, always carefully consider the left and right limits when finding derivatives!

Consider the function f(x) = x|x|. Find a formula for the slope of the tangent line to this curve at the point (a, a|a|) considering three separate cases: when a > 0, when a < 0, and when a = 0.

Learn how to find the slope of the tangent line to the function f(x) = x|x| at various points. This problem covers key calculus concepts, including derivatives of piecewise functions and their applications. Suitable for students in Grades 11-12.

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