https://file.aiquickdraw.com/mathbot/file/1727235905985zp4gwbtq.jpg

Let's analyze the problem shown in the image.

### Given Information:
You are provided with two functions \( f(x) \) and \( g(x) \) whose graphs are shown. Two new functions are defined:
- \( u(x) = f(x)g(x) \)
- \( v(x) = \frac{f(x)}{g(x)} \)

You are asked to compute:
1. \( u'(1) \) (which you got correct).
2. \( v'(5) \) (for which you received incorrect feedback).

### Strategy:
To solve for these derivatives, we will need the **Product Rule** and the **Quotient Rule**. 

- **Product Rule**: 
   \[
   u'(x) = f'(x)g(x) + f(x)g'(x)
   \]

- **Quotient Rule**: 
   \[
   v'(x) = \frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2}
   \]

### Part (b) \( v'(5) \):

To compute \( v'(5) = \frac{f'(5)g(5) - f(5)g'(5)}{[g(5)]^2} \), we need to:
1. Find the values of \( f(5) \) and \( g(5) \) from the graph.
2. Find the slopes \( f'(5) \) and \( g'(5) \), which are the derivatives of \( f(x) \) and \( g(x) \) at \( x = 5 \).

From the graph:
- \( f(5) \) and \( g(5) \) are the \( y \)-values at \( x = 5 \).
- \( f'(5) \) and \( g'(5) \) are the slopes at \( x = 5 \).

Let me analyze the graph more closely to extract the required values.I've opened the image for closer inspection. Now, let's extract the relevant values from the graph at \( x = 5 \):

1. **\( f(5) \) and \( g(5) \)**:
   - From the graph, find the y-values of \( f \) and \( g \) at \( x = 5 \).

2. **\( f'(5) \) and \( g'(5) \)**:
   - These are the slopes of the lines at \( x = 5 \), calculated by selecting two points on the line around \( x = 5 \) and applying the slope formula:
   \[
   \text{slope} = \frac{y_2 - y_1}{x_2 - x_1}
   \]

Let me interpret the graph visually to find these values.Upon closer inspection of the graph:

- **At \( x = 5 \)**:
  - \( f(5) = 2 \)
  - \( g(5) = 4 \)

- Now for the slopes at \( x = 5 \):
  - **\( f'(5) \)** (slope of \( f(x) \) near \( x = 5 \)): 
    Using two points from the graph for \( f(x) \) between \( x = 4 \) and \( x = 6 \):
    - The points seem to be \( (4, 0) \) and \( (6, 4) \), giving the slope:
    \[
    f'(5) = \frac{4 - 0}{6 - 4} = \frac{4}{2} = 2
    \]

  - **\( g'(5) \)** (slope of \( g(x) \) near \( x = 5 \)): 
    Using points around \( x = 5 \) for \( g(x) \):
    - Points seem to be \( (4, 3) \) and \( (6, 5) \), giving the slope:
    \[
    g'(5) = \frac{5 - 3}{6 - 4} = \frac{2}{2} = 1
    \]

Now let's plug these values into the **Quotient Rule** for \( v'(5) \):
\[
v'(5) = \frac{f'(5)g(5) - f(5)g'(5)}{[g(5)]^2}
\]
\[
v'(5) = \frac{2 \cdot 4 - 2 \cdot 1}{4^2} = \frac{8 - 2}{16} = \frac{6}{16} = 0.375
\]

Thus, \( v'(5) = 0.375 \).

Would you like to clarify any details or have further questions? Here are some related questions for practice:

1. How do you apply the product rule to \( u'(5) \)?
2. How do you calculate the slope of a line from two points?
3. Can the quotient rule be used when the denominator equals zero?
4. How does the chain rule relate to product and quotient rules?
5. Why does \( g(5) \) play a key role in the quotient rule formula?

**Tip**: Always ensure you accurately extract function values and slopes from graphs to apply derivative rules correctly.

Find u'(1) and v'(5) where u(x) = f(x)g(x) and v(x) = f(x)/g(x), given graphs of f(x) and g(x).

Learn how to compute the derivatives u'(1) and v'(5) using the product and quotient rules, with values extracted from graphs. This solution focuses on slope calculations, and applying the product and quotient rules for differentiating functions.

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