https://file.aiquickdraw.com/mathbot/file/1727724004027mcdvf5mi.jpg

Let's analyze the problem step by step based on the given graph and information.

The question is asking to find derivatives of the functions defined by:
- \( u(x) = f(g(x)) \),
- \( v(x) = g(f(x)) \),
- \( w(x) = g(g(x)) \).

### (a) Finding \( u'(1) = \frac{d}{dx}[f(g(x))] \) at \( x = 1 \)
By the chain rule, the derivative of \( u(x) \) is:
\[
u'(x) = f'(g(x)) \cdot g'(x)
\]
Now, let's plug in \( x = 1 \):
\[
u'(1) = f'(g(1)) \cdot g'(1)
\]
From the graph:
- \( g(1) = 5 \) (from the blue curve at \( x = 1 \)).
- We now need \( f'(5) \) and \( g'(1) \):
  - \( f'(5) \): At \( x = 5 \), the red graph (representing \( f(x) \)) has a sharp corner, indicating that the derivative does not exist at \( x = 5 \).
  - \( g'(1) \): From the blue graph at \( x = 1 \), the slope appears to be 0, as it is a flat horizontal line.

Thus, \( u'(1) \) does **not** exist because \( f'(5) \) does not exist.

### (b) Finding \( v'(1) = \frac{d}{dx}[g(f(x))] \) at \( x = 1 \)
Using the chain rule again:
\[
v'(x) = g'(f(x)) \cdot f'(x)
\]
Now, evaluate at \( x = 1 \):
\[
v'(1) = g'(f(1)) \cdot f'(1)
\]
From the graph:
- \( f(1) = 1 \) (from the red curve at \( x = 1 \)),
- Therefore, we need \( g'(1) \) and \( f'(1) \):
  - \( g'(1) = 0 \) (as previously noted, the blue graph is flat at \( x = 1 \)).
  - \( f'(1) \): At \( x = 1 \), the red graph has a positive slope of 1.

Thus, since \( g'(1) = 0 \), \( v'(1) = 0 \).

### (c) Finding \( w'(1) = \frac{d}{dx}[g(g(x))] \) at \( x = 1 \)
Again, by the chain rule:
\[
w'(x) = g'(g(x)) \cdot g'(x)
\]
At \( x = 1 \):
\[
w'(1) = g'(g(1)) \cdot g'(1)
\]
From the graph:
- \( g(1) = 5 \),
- We need \( g'(5) \) and \( g'(1) \):
  - \( g'(1) = 0 \) (flat blue curve at \( x = 1 \)),
  - \( g'(5) \): The blue graph has a sharp corner at \( x = 5 \), so the derivative \( g'(5) \) does not exist.

Thus, \( w'(1) \) does **not** exist because \( g'(5) \) does not exist.

### Final Answers:
- (a) \( u'(1) \) does not exist because \( f'(5) \) does not exist.
- (b) \( v'(1) = 0 \).
- (c) \( w'(1) \) does not exist because \( g'(5) \) does not exist.

Let me know if you want further details on any part of the solution! 

Here are 5 related questions:
1. How is the chain rule applied to composite functions like \( f(g(x)) \)?
2. What does it mean for a derivative to not exist at a corner?
3. How can we estimate the slope from a graph visually?
4. Why does \( g'(1) = 0 \) despite the function being defined?
5. Can you describe a situation where a derivative exists but equals zero?

**Tip:** Sharp corners in a function's graph are key indicators that the derivative does not exist at that 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.

Learn how to find the derivatives of composite functions u(x) = f(g(x)), v(x) = g(f(x)), and w(x) = g(g(x)) using the chain rule. This solution involves interpreting graphs of the functions and explaining where the derivatives do not exist due to sharp corners or undefined slopes.

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