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.

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.

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!

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.