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The given function is \( f(x) = 2 + |x - 3| \), and you are asked to find the value of the derivative \( f'(x) \) at \( x = 3 \).

### Key points:
- The absolute value function \( |x - 3| \) introduces a potential non-differentiability at \( x = 3 \).
- We will first check if the function is differentiable at \( x = 3 \).

### Step-by-Step Analysis:

1. **Piecewise Definition of the Function**:
   The function can be written as:
   \[
   f(x) = 2 + \begin{cases} 
   x - 3 & \text{if } x \geq 3 \\
   3 - x & \text{if } x < 3
   \end{cases}
   \]

2. **Derivative on Either Side of \( x = 3 \)**:
   - For \( x > 3 \), \( f(x) = 2 + (x - 3) = x - 1 \), and its derivative is \( f'(x) = 1 \).
   - For \( x < 3 \), \( f(x) = 2 + (3 - x) = 5 - x \), and its derivative is \( f'(x) = -1 \).

3. **At \( x = 3 \)**:
   - As we approach from the right (\( x \to 3^+ \)), \( f'(x) = 1 \).
   - As we approach from the left (\( x \to 3^- \)), \( f'(x) = -1 \).
   - Since the left-hand and right-hand derivatives are not equal, the function is not differentiable at \( x = 3 \).

### Conclusion:
The derivative does not exist at \( x = 3 \).

The correct answer is **(E) Nonexistent**.

Would you like further explanation on any of the steps?

Here are five related questions:
1. What does it mean for a function to be differentiable at a point?
2. How do you compute the derivative of an absolute value function?
3. What is the significance of the left-hand and right-hand derivatives in determining differentiability?
4. How do piecewise functions affect differentiability and continuity?
5. Can you provide more examples of functions that are not differentiable at a point?

**Tip**: Always check both continuity and the equality of the left-hand and right-hand derivatives when dealing with absolute value functions at critical points.

If f(x) = 2 + |x - 3| for all x, then the value of the derivative f'(x) at x = 3 is:

Learn how to find the derivative of the function f(x) = 2 + |x - 3| at x = 3, covering piecewise functions and differentiation techniques. Explore the non-differentiability at sharp corners using left-hand and right-hand derivatives. Suitable for high school calculus and college-level students.

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