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The problem asks to determine the derivatives of composite functions \( u(x) = f(g(x)) \), \( v(x) = g(f(x)) \), and \( w(x) = g(g(x)) \) at \( x = 1 \), using the given graphs of \( f(x) \) and \( g(x) \).

### Analyzing the functions and their derivatives:
1. **For \( u'(1) = \frac{d}{dx}[f(g(x))] \):**
   - By the chain rule, \( u'(x) = f'(g(x)) \cdot g'(x) \).
   - To find \( u'(1) \), we need both \( f'(g(1)) \) and \( g'(1) \).
   - From the graph, \( g(1) = 3 \), and we need to check \( f'(3) \) and \( g'(1) \).
     - The graph shows a sharp corner at \( g(x) = 1 \), indicating that \( g'(1) \) does not exist.

   **Conclusion:** \( u'(1) \) does not exist because \( g'(1) \) does not exist.

2. **For \( v'(1) = \frac{d}{dx}[g(f(x))] \):**
   - By the chain rule, \( v'(x) = g'(f(x)) \cdot f'(x) \).
   - To find \( v'(1) \), we need both \( g'(f(1)) \) and \( f'(1) \).
   - From the graph, \( f(1) = 2 \), so we need to evaluate \( g'(2) \) and \( f'(1) \).
     - The graph shows that \( g'(2) \) exists and appears to be 1 (since the slope at \( x = 2 \) for \( g(x) \) is 1).
     - The graph shows \( f'(1) = 0 \) (as the slope of \( f(x) \) at \( x = 1 \) is 0).

   **Conclusion:** \( v'(1) = g'(f(1)) \cdot f'(1) = g'(2) \cdot 0 = 0 \).

3. **For \( w'(1) = \frac{d}{dx}[g(g(x))] \):**
   - By the chain rule, \( w'(x) = g'(g(x)) \cdot g'(x) \).
   - To find \( w'(1) \), we need both \( g'(g(1)) \) and \( g'(1) \).
   - From the graph, \( g(1) = 3 \), and we need to check \( g'(3) \) and \( g'(1) \).
     - The graph shows that \( g'(1) \) does not exist, as there is a sharp corner at \( x = 1 \).

   **Conclusion:** \( w'(1) \) does not exist because \( g'(1) \) does not exist.

### Summary:
- (a) \( u'(1) \) does not exist because \( g'(1) \) does not exist.
- (b) \( v'(1) = 0 \).
- (c) \( w'(1) \) does not exist because \( g'(1) \) does not exist.

Would you like more details on any step? Here are some related questions for deeper understanding:

1. How does the chain rule apply to composite functions like \( u(x) \), \( v(x) \), and \( w(x) \)?
2. What is the significance of a sharp corner in the graph in terms of the derivative?
3. Can you explain how to visually identify if a function is differentiable at a certain point on a graph?
4. How can one interpret the slopes of piecewise linear functions like \( f(x) \) and \( g(x) \)?
5. Why does \( v'(1) \) equal zero even though \( g'(2) \neq 0 \)?

**Tip:** Always check for sharp corners or discontinuities on a graph to determine if the derivative exists at a specific point.

If f and g are the functions whose graphs are shown, let u(x) = f(g(x)), v(x) = g(f(x)), and w(x) = g(g(x)). Find each derivative, if it exists. If it does not exist, explain why. (If an answer does not exist, enter DNE.)

This solution explores how to apply the chain rule to find the derivatives of composite functions u(x) = f(g(x)), v(x) = g(f(x)), and w(x) = g(g(x)). It also addresses cases where derivatives do not exist due to sharp corners in graphs, with detailed step-by-step reasoning.

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