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We are asked to find the derivative of the function \( f(x) = \frac{g(x)}{h(x)} \) at \( x = 3 \), denoted as \( f'(3) \). 

### Step 1: Apply the Quotient Rule
The derivative of a quotient of two functions, \( f(x) = \frac{g(x)}{h(x)} \), is given by the quotient rule:

\[
f'(x) = \frac{g'(x)h(x) - g(x)h'(x)}{(h(x))^2}
\]

We will apply this rule to find \( f'(3) \).

### Step 2: Find \( g(3), g'(3), h(3), \) and \( h'(3) \) from the graphs.

1. **For \( g(x) \):**
   - From the graph, at \( x = 3 \), we can see that \( g(3) = 3 \).
   - The slope of the line \( g(x) \) is constant (it's a straight line), so we can calculate the slope between two points, say \( (1, 7) \) and \( (4, 2) \). The slope is:
   
   \[
   g'(x) = \frac{7 - 2}{1 - 4} = \frac{5}{-3} = -\frac{5}{3}
   \]

   Hence, \( g'(3) = -\frac{5}{3} \).

2. **For \( h(x) \):**
   - From the graph, at \( x = 3 \), we see that \( h(3) = 5 \).
   - To find \( h'(3) \), notice that \( h(x) \) has a vertex at \( x = 3 \), which means the slope \( h'(3) = 0 \) because the graph is flat at this point.

### Step 3: Apply the values to the Quotient Rule

Now that we have all the values:
- \( g(3) = 3 \)
- \( g'(3) = -\frac{5}{3} \)
- \( h(3) = 5 \)
- \( h'(3) = 0 \)

Substitute these into the quotient rule:

\[
f'(3) = \frac{\left(-\frac{5}{3}\right)(5) - (3)(0)}{(5)^2} = \frac{-\frac{25}{3}}{25} = -\frac{1}{3}
\]

### Final Answer:
\[
f'(3) = -\frac{1}{3}
\]

Do you want any further clarifications or details? Here are 5 related questions you might want to explore:
1. What is the general form of the quotient rule for derivatives?
2. How do you find the slope of a line from a graph?
3. What happens when the derivative at a point is zero?
4. Can the quotient rule be applied to functions that are not linear?
5. What are the limitations of the quotient rule when the denominator is zero?

**Tip:** When applying the quotient rule, ensure that both the numerator and denominator are differentiable at the point of interest to avoid complications.

If f(x) = g(x) / h(x), find f'(3) using the provided graphs for g(x) and h(x).

Learn how to find the derivative of the quotient function f(x) = g(x) / h(x) at x = 3 using the quotient rule. Analyze the graphs of g(x) and h(x) to determine values such as g(3), g'(3), h(3), and h'(3). Suitable for high school calculus students in Grades 11-12.

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