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To find the derivative of the product of two functions \( f(x) \) and \( g(x) \) at \( x = 0 \), we will use the **product rule** for derivatives. The product rule states:

\[
\frac{d}{dx} [f(x)g(x)] = f'(x)g(x) + f(x)g'(x)
\]

We need to evaluate this expression at \( x = 0 \). This requires two things:

1. The values of \( f(x) \) and \( g(x) \) at \( x = 0 \).
2. The slopes (derivatives) \( f'(x) \) and \( g'(x) \) at \( x = 0 \).

### From the graph:
- **\( f(x) \)** is the black line, and **\( g(x) \)** is the blue line.

#### Step 1: Evaluate the functions at \( x = 0 \):
- \( f(0) \): From the graph, the black line (representing \( f(x) \)) has a value of \( 2 \) at \( x = 0 \), so \( f(0) = 2 \).
- \( g(0) \): The blue line (representing \( g(x) \)) has a value of \( 6 \) at \( x = 0 \), so \( g(0) = 6 \).

#### Step 2: Find the slopes at \( x = 0 \):
- The slope of \( f(x) \) is constant because it is a straight line. We can calculate it by picking two points on the line:
  - At \( x = 0 \), \( f(0) = 2 \).
  - At \( x = 6 \), \( f(6) = 6 \).
  - The slope \( f'(x) = \frac{6 - 2}{6 - 0} = \frac{4}{6} = \frac{2}{3} \).

- Similarly, the slope of \( g(x) \) is constant. From the graph:
  - At \( x = 0 \), \( g(0) = 6 \).
  - At \( x = 6 \), \( g(6) = 0 \).
  - The slope \( g'(x) = \frac{0 - 6}{6 - 0} = -1 \).

### Step 3: Apply the product rule:
Now substitute these values into the product rule formula:
\[
\frac{d}{dx} [f(x)g(x)] \bigg|_{x=0} = f'(0)g(0) + f(0)g'(0)
\]
\[
= \left( \frac{2}{3} \times 6 \right) + (2 \times -1)
\]
\[
= 4 - 2 = 2
\]

### Final Answer:
\[
\frac{d}{dx} [f(x)g(x)] \bigg|_{x=0} = 2
\]

Would you like further clarification or more details? Here are five related questions to deepen your understanding:

1. How is the product rule derived from the limit definition of a derivative?
2. What is the geometrical interpretation of the product rule?
3. How would the result change if one of the functions was constant?
4. What would the derivative be if we shifted the evaluation point to \( x = 2 \)?
5. How can the product rule be extended to more than two functions?

**Tip:** When applying the product rule, always check the slopes and values of the functions at the given point carefully from the graph.

Use the figure to find the following derivative: d/dx [f(x)g(x)] |x=0, where f(x) is the black line and g(x) is the blue line.

Learn how to compute the derivative of the product of two functions f(x) and g(x) using the product rule. This example covers reading values and slopes from a graph to solve the problem at x = 0. Suitable for high school students in Grades 11-12 studying calculus.

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