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To solve for \( h'(3) \) given \( h(x) = f(x)g(x) \), we need to use the product rule for differentiation:

\[
h'(x) = f'(x)g(x) + f(x)g'(x)
\]

Our task is to find \( h'(3) \), which means evaluating \( h'(x) \) at \( x = 3 \).

### Step 1: Identify \( f(x) \) and \( g(x) \) at \( x = 3 \)

From the graph:
- \( f(3) \) appears to be around \( 4 \).
- \( g(3) \) appears to be around \( 2 \).

### Step 2: Determine \( f'(3) \) and \( g'(3) \)

#### For \( f(x) \):
The function \( f(x) \) is linear, so it has a constant slope. Observing the line, \( f(x) \) has a negative slope. We can calculate the slope by choosing two points on the line:
- At \( x = 0 \), \( f(x) = 7 \).
- At \( x = 6 \), \( f(x) = 0 \).

The slope of \( f(x) \), or \( f'(x) \), is:
\[
f'(x) = \frac{0 - 7}{6 - 0} = -\frac{7}{6}
\]

Thus, \( f'(3) = -\frac{7}{6} \).

#### For \( g(x) \):
The function \( g(x) \) is piecewise linear. At \( x = 3 \), we need the slope of \( g(x) \) in the interval around \( x = 3 \). Observing the line segments, between \( x = 2 \) and \( x = 4 \):
- At \( x = 2 \), \( g(x) = 1 \).
- At \( x = 4 \), \( g(x) = 3 \).

The slope of \( g(x) \), or \( g'(x) \), in this interval is:
\[
g'(x) = \frac{3 - 1}{4 - 2} = 1
\]

Thus, \( g'(3) = 1 \).

### Step 3: Apply the Product Rule

Now we substitute these values into the product rule formula:
\[
h'(3) = f'(3)g(3) + f(3)g'(3)
\]

Substituting the values:
\[
h'(3) = \left(-\frac{7}{6}\right)(2) + (4)(1)
\]

Calculating each term:
1. \( f'(3)g(3) = -\frac{7}{6} \times 2 = -\frac{7}{3} \)
2. \( f(3)g'(3) = 4 \times 1 = 4 \)

Adding these:
\[
h'(3) = -\frac{7}{3} + 4 = -\frac{7}{3} + \frac{12}{3} = \frac{5}{3}
\]

### Final Answer

\[
h'(3) = \frac{5}{3}
\]

Would you like further details or have any questions?

---

1. How is the product rule used in this problem?
2. Why is it necessary to calculate the slopes \( f'(3) \) and \( g'(3) \)?
3. How do we interpret a piecewise function in derivative calculations?
4. What are the implications if \( g(x) \) were not piecewise?
5. Could this problem be solved using another method?

**Tip:** When dealing with piecewise functions, always check the interval around the point of interest to determine the correct slope.

The graphs of the linear function f and the piecewise linear function g are shown in the figure above. If h(x) = f(x)g(x), then h'(3) = ?

Learn how to solve for h'(3) when h(x) = f(x)g(x), using the product rule for differentiation. The problem includes analyzing linear and piecewise functions from a graph, calculating slopes, and applying derivative rules. Ideal for advanced high school or early college students.

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