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To solve this problem, we first need to understand the function \( f(x) = |x| \) and its derivative.

### Step 1: Understanding the function \( f(x) = |x| \)
The function \( f(x) = |x| \) is defined as:
\[
f(x) = 
\begin{cases} 
x & \text{if } x \geq 0, \\
-x & \text{if } x < 0.
\end{cases}
\]

### Step 2: Derivative of \( f(x) = |x| \)
The derivative of \( f(x) \) with respect to \( x \) is piecewise defined as well:
\[
f'(x) = 
\begin{cases} 
1 & \text{if } x > 0, \\
-1 & \text{if } x < 0, \\
\text{undefined} & \text{if } x = 0.
\end{cases}
\]
Note that at \( x = 0 \), the derivative does not exist because there is a sharp corner (cusp) at \( x = 0 \) in the graph of \( |x| \).

### Step 3: Analyze the options
Now, let's analyze each of the options provided:

- **(a) \( f'(x) = 1 \) if \( x > 0 \)**: This is true since the derivative of \( |x| \) is \( 1 \) for positive \( x \).

- **(b) \( f'(x) = -1 \) if \( x > 0 \)**: This is false. As noted above, \( f'(x) = 1 \) when \( x > 0 \), not \(-1\).

- **(c) \( f'(x) = 0 \) if \( x > 0 \)**: This is false. The correct derivative for \( x > 0 \) is \( 1 \), not \( 0 \).

- **(d) \( f'(x) = 1 \) if \( x < 0 \)**: This is false. For \( x < 0 \), the derivative is \( -1 \).

### Conclusion:
The correct answer is **(a) \( f'(x) = 1 \) if \( x > 0 \)**.

Would you like further details or have any questions? Here are some additional questions related to this topic:

1. What is the derivative of \( |x| \) at \( x = 0 \)?
2. How do we find the derivative of a piecewise function?
3. What does the graph of \( |x| \) look like?
4. What is a cusp in a function's graph, and how does it affect differentiability?
5. How can you determine the derivative of an absolute value function without calculus?
6. Can a function be continuous but not differentiable? Provide an example.
7. How does the concept of the derivative apply to real-world problems involving absolute values?
8. What are other examples of functions that are not differentiable at certain points?

**Tip:** When dealing with absolute value functions, always consider the piecewise definition for analysis and derivative calculation.

Explore the derivative of the absolute value function |x| with a detailed explanation and step-by-step analysis. Learn about the piecewise definition of the function, its derivative at different points, and how to apply calculus concepts to solve related problems. Suitable for high school students studying calculus and derivatives.

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