https://file.aiquickdraw.com/mathbot/file/17289290299453gwb1fvn.jpg

Let's walk through the solution based on what the image shows.

### Functions Provided
- \( F(x) \): represented by the **blue graph** (left panel).
- \( G(x) \): represented by the **red graph** (right panel).

We need to calculate the derivatives for two new functions defined as:
- \( P(x) = F(x)G(x) \)
- \( Q(x) = \frac{F(x)}{G(x)} \)

### Part (a): \( P'(6) \)
This is already given in the answer box as **6**.

To calculate it yourself, use the **product rule**:
\[
P'(x) = F'(x)G(x) + F(x)G'(x)
\]
At \(x = 6\):
1. From the blue graph, find \( F(6) \) and \( F'(6) \).
2. From the red graph, find \( G(6) \) and \( G'(6) \).

You’d substitute those values into the product rule formula.

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### Part (b): \( Q'(1) \)
We need to compute this value using the **quotient rule**:
\[
Q'(x) = \frac{F'(x)G(x) - F(x)G'(x)}{[G(x)]^2}
\]
At \(x = 1\):
1. From the blue graph, find \( F(1) \) and \( F'(1) \).
2. From the red graph, find \( G(1) \) and \( G'(1) \).

#### Now:
- Substitute these values into the formula for the quotient rule.
- Ensure that the derivative values match what you read off the graphs.

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If you can provide the key values from the graphs at \(x = 1\) (for both functions and their slopes), I can help complete the answer!

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Would you like help interpreting the graphs more closely? Here are some follow-up questions:

1. What are the values of \(F(1)\) and \(G(1)\) from the graphs?
2. Can you estimate the slopes \(F'(1)\) and \(G'(1)\)?
3. Do you understand how the quotient rule and product rule were applied?
4. Would you like to double-check if the answer \(P'(6) = 6\) was correctly derived?
5. Are you comfortable reading the slopes from the graphs at specific points?

**Tip:** Remember, the derivative at a point is the slope of the tangent line at that point on the function’s graph.

Calculate the derivatives of P(x) = F(x)G(x) and Q(x) = F(x)/G(x) using the product and quotient rules, based on the graphs of the functions F (blue) and G (red). Part (a) asks for P'(6), and part (b) asks for Q'(1).

Learn how to calculate derivatives using the product and quotient rules based on the graphs of F and G. This problem involves applying derivative rules and interpreting graphs to find key values. Suitable for students in Grades 11-12 studying Calculus I.

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