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.