https://file.aiquickdraw.com/mathbot/file/17303533090552ey1w4i4.jpg

To solve the problem, let's analyze each part carefully using the information given on the graph.

Given:
- \( h(x) = \frac{f(x)}{g(x)} \)
- We are asked to find the derivatives \( h'(2) \), \( h'(3) \), and \( h'(4) \) using the graph.

The derivative \( h'(x) \) of a quotient function \( h(x) = \frac{f(x)}{g(x)} \) is given by the quotient rule:
\[
h'(x) = \frac{f'(x) g(x) - f(x) g'(x)}{[g(x)]^2}
\]
where \( f'(x) \) and \( g'(x) \) are the derivatives of \( f(x) \) and \( g(x) \), respectively.

Let's examine each part:

### Part (a): Find \( h'(2) \)
- **Solution:** From the graph, we determine \( f(2) \), \( g(2) \), \( f'(2) \), and \( g'(2) \).
  - \( f(2) = 2 \)
  - \( g(2) = 1 \)
  - \( f'(2) \approx -1 \) (since \( f(x) \) has a slope of \(-1\) at \( x = 2 \))
  - \( g'(2) = 0 \) (since \( g(x) \) is flat at \( x = 2 \))
  - Using the quotient rule:
    \[
    h'(2) = \frac{f'(2) g(2) - f(2) g'(2)}{[g(2)]^2} = \frac{(-1)(1) - (2)(0)}{1^2} = -1
    \]
  - **Answer:** \( h'(2) = -1 \) (This matches the correct answer already marked.)

### Part (b): Find \( h'(3) \)
- **Solution:** From the graph, we find \( f(3) \), \( g(3) \), \( f'(3) \), and \( g'(3) \).
  - \( f(3) = 0 \)
  - \( g(3) = 0 \) (since both functions intersect the x-axis at \( x = 3 \))
  - Because both \( f(3) = 0 \) and \( g(3) = 0 \), the function \( h(x) = \frac{f(x)}{g(x)} \) is undefined at \( x = 3 \), so \( h'(3) \) does not exist.
  - **Answer:** \( h'(3) = \text{DNE} \) (This also matches the correct answer.)

### Part (c): Find \( h'(4) \)
- **Solution:** Let's determine \( f(4) \), \( g(4) \), \( f'(4) \), and \( g'(4) \).
  - \( f(4) = 1 \)
  - \( g(4) = 2 \)
  - \( f'(4) = 1 \) (since \( f(x) \) has a slope of \( 1 \) at \( x = 4 \))
  - \( g'(4) = 0 \) (since \( g(x) \) is flat at \( x = 4 \))
  - Applying the quotient rule:
    \[
    h'(4) = \frac{f'(4) g(4) - f(4) g'(4)}{[g(4)]^2} = \frac{(1)(2) - (1)(0)}{2^2} = \frac{2}{4} = \frac{1}{2}
    \]
  - **Answer:** \( h'(4) = \frac{1}{2} \)

Let me know if you'd like further explanation on any part of the solution or if you have additional questions!

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### Related Questions:
1. How is the quotient rule applied to find derivatives of function quotients?
2. What happens if both the numerator and denominator equal zero at a given point?
3. How do we interpret the slopes of piecewise functions from a graph?
4. How would \( h(x) \) change if \( g(x) \) were to become negative?
5. What is the significance of a point where \( h(x) \) is undefined in a derivative context?

**Tip:** When applying the quotient rule, always check for cases where the denominator is zero, as this could indicate an undefined point or require careful handling to avoid division by zero.

Use the figure to find the indicated derivatives, if they exist. Let h(x) = f(x) / g(x). Find h'(2), h'(3), and h'(4).

Learn how to find derivatives of the function h(x) = f(x) / g(x) using the quotient rule and graphical data. This problem includes calculating h'(2), h'(3), and h'(4) by analyzing slopes and values from the graph. Suitable for college calculus students or advanced high school students.

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