To differentiate the absolute value function \( |x| \), you need to consider the definition of the absolute value and treat it piecewise. The absolute value function is defined as:

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

### 1. Piecewise differentiation
Based on this definition, the derivative of \( |x| \) is also piecewise:

\[
\frac{d}{dx} |x| = 
\begin{cases} 
1 & \text{if } x > 0 \\
-1 & \text{if } x < 0 \\
\text{undefined} & \text{if } x = 0
\end{cases}
\]

### 2. Analyzing the derivative at \( x = 0 \)
At \( x = 0 \), the derivative is not defined because the left-hand and right-hand derivatives are different:

- For \( x > 0 \), \( \frac{d}{dx} |x| = 1 \).
- For \( x < 0 \), \( \frac{d}{dx} |x| = -1 \).

The sudden change in slope at \( x = 0 \) means the derivative does not exist at that point.

### 3. General form using the sign function
Alternatively, the derivative of \( |x| \) can be expressed using the sign function \( \text{sgn}(x) \), which is defined as:

\[
\text{sgn}(x) = 
\begin{cases} 
1 & \text{if } x > 0 \\
0 & \text{if } x = 0 \\
-1 & \text{if } x < 0 
\end{cases}
\]

So the derivative can be written as:

\[
\frac{d}{dx} |x| = \text{sgn}(x) \text{ for } x \neq 0
\]

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Would you like to discuss this further or explore any additional questions related to derivatives?

Here are some related questions:

1. How do you differentiate functions involving absolute values, such as \( |f(x)| \)?
2. What is the graphical interpretation of the derivative of the absolute value function?
3. Can you integrate absolute value functions, and how is it done?
4. How do you handle absolute value functions in optimization problems?
5. What are some applications of the derivative of absolute value in physics or economics?

**Tip:** When working with piecewise functions like the absolute value, always consider the points where the function changes its form (e.g., at \( x = 0 \) for \( |x| \)). These are often the points where the derivative might be undefined or behave differently.

Learn how to differentiate the absolute value function |x| using piecewise differentiation and the sign function. Understand the definition and behavior of the derivative at different points. Suitable for undergraduate students studying calculus and piecewise functions.

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