https://file.aiquickdraw.com/mathbot/file/1730335954493r51xfych.jpg

Let's analyze and solve each part of this problem based on the given information.

1. **Identifying \( u(x) = f(x) \cdot g(x) \) and finding \( u'(x) \):**
   Since \( u(x) = f(x) \cdot g(x) \), we use the product rule to find \( u'(x) \):
   \[
   u'(x) = f'(x) \cdot g(x) + f(x) \cdot g'(x)
   \]
   For \( u'(1) \), we need the values of \( f(1) \), \( g(1) \), \( f'(1) \), and \( g'(1) \) from the graph.

2. **Identifying \( v(x) = \frac{f(x)}{g(x)} \) and finding \( v'(x) \):**
   Since \( v(x) = \frac{f(x)}{g(x)} \), we use the quotient rule to find \( v'(x) \):
   \[
   v'(x) = \frac{f'(x) \cdot g(x) - f(x) \cdot g'(x)}{[g(x)]^2}
   \]
   For \( v'(5) \), we need the values of \( f(5) \), \( g(5) \), \( f'(5) \), and \( g'(5) \) from the graph.

### Step-by-Step Solutions

#### (a) Find \( u'(1) \):
From the graph:
- \( f(1) = 3 \)
- \( g(1) = 1 \)
- \( f'(1) \): The slope of \( f(x) \) at \( x = 1 \) is the slope of the line segment around \( x = 1 \), which appears to be \( -1 \).
- \( g'(1) \): The slope of \( g(x) \) at \( x = 1 \) is also calculated from the slope of the segment, which appears to be \( 2 \).

Now, plug these values into the formula for \( u'(x) \):
\[
u'(1) = f'(1) \cdot g(1) + f(1) \cdot g'(1) = (-1) \cdot 1 + 3 \cdot 2 = -1 + 6 = 5
\]
So, \( u'(1) = 5 \).

#### (b) Find \( v'(5) \):
From the graph:
- \( f(5) = 2 \)
- \( g(5) = 3 \)
- \( f'(5) \): The slope of \( f(x) \) at \( x = 5 \) is calculated from the nearby segment, which appears to be \( -1 \).
- \( g'(5) \): The slope of \( g(x) \) at \( x = 5 \) is also calculated from the segment, which appears to be \( 1 \).

Now, plug these values into the formula for \( v'(x) \):
\[
v'(5) = \frac{f'(5) \cdot g(5) - f(5) \cdot g'(5)}{[g(5)]^2} = \frac{(-1) \cdot 3 - 2 \cdot 1}{3^2} = \frac{-3 - 2}{9} = \frac{-5}{9}
\]
So, \( v'(5) = -\frac{5}{9} \).

### Answers
- (a) \( u'(1) = 5 \)
- (b) \( v'(5) = -\frac{5}{9} \)

Would you like more details, or have questions on any step? 

#### Additional Related Questions:
1. How would the result change if \( f(x) \) or \( g(x) \) had a different slope at \( x = 1 \)?
2. What if we needed to find \( u'(x) \) for another value, such as \( x = 2 \)?
3. How would the derivative change if \( u(x) \) were defined as \( f(x) + g(x) \) instead of \( f(x) \cdot g(x) \)?
4. What are the implications if \( g(x) \) had a value of zero at \( x = 1 \)?
5. Could we find \( v'(x) \) if \( g(x) \) were zero at \( x = 5 \)? What would happen?

#### Tip:
Always verify the slopes (derivatives) carefully by calculating the rise-over-run for each segment on the graph; it ensures precise differentiation.

Let u(x) = f(x)g(x) and v(x) = f(x)/g(x). Find u'(1) and v'(5) based on the graphs of functions f and g.

Learn how to calculate the derivatives u'(1) and v'(5) for functions u(x) = f(x)g(x) and v(x) = f(x)/g(x) using the product and quotient rules. This problem involves understanding derivatives through graphical interpretation and applying calculus concepts. Suitable for Grades 11-12 or introductory calculus.

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