Let's address this problem step by step. We are given two functions, $F(x)$ and $G(x)$, and we need to find derivatives of the composite products $P(x) = F(x)G(x)$ and $Q(x) = \frac{F(x)}{G(x)}$.

To solve the problem:

(a) Find $P'(2)$

Given that $P(x) = F(x)G(x)$, we use the product rule for differentiation:

$P'(x) = F'(x)G(x) + F(x)G'(x)$

We are tasked with finding $P'(2)$, so we need to evaluate $F(2), G(2), F'(2),$ and $G'(2)$.

From the graph descriptions:

• $F(2) = 3$ (since the function passes through $(2, 3)$).
• $G(2) = 2$ (since the function passes through $(2, 2)$).

Now, to estimate the derivatives $F'(2)$ and $G'(2)$, we need to use the graph's behavior around $x = 2$:

• $F'(2)$ can be approximated by the slope of the tangent line at $x = 2$. The graph indicates that the function changes direction at $(2, 3)$, suggesting the slope is zero at this point. Hence, $F'(2) = 0$.
• $G'(2)$ can be calculated by the slope of the line segment between $(0, 1)$ and $(2, 2)$, which gives a slope of:

$G'(2) = \frac{2 - 1}{2 - 0} = \frac{1}{2}$

Using these values in the product rule:

$P'(2) = F'(2)G(2) + F(2)G'(2) = (0)(2) + (3)\left( \frac{1}{2} \right) = 0 + \frac{3}{2} = \frac{3}{2}$

Thus, the correct value is:

$P'(2) = \frac{3}{2}$

(b) Find $Q'(7)$

Given that $Q(x) = \frac{F(x)}{G(x)}$, we use the quotient rule for differentiation:

$Q'(x) = \frac{F'(x)G(x) - F(x)G'(x)}{(G(x))^2}$

We are tasked with finding $Q'(7)$, so we need to evaluate $F(7), G(7), F'(7),$ and $G'(7)$.

From the graph descriptions:

• $F(7) = 5$ (since the function passes through $(7, 5)$).
• $G(7) = 1$ (since the function passes through $(7, 1)$).

Now, we estimate the derivatives $F'(7)$ and $G'(7)$:

• $F'(7)$ can be approximated by the slope of the line segment around $x = 7$. Based on the graph description, the function is increasing linearly between $x = 3$ and $x = 7$, so we can calculate the slope:

$F'(7) = \frac{5 - 4}{7 - 3} = \frac{1}{4}$

• $G'(7)$ can be calculated by the slope of the line segment between $(4, 3)$ and $(7, 1)$, which gives a slope of:

$G'(7) = \frac{1 - 3}{7 - 4} = \frac{-2}{3}$

Using these values in the quotient rule:

$Q'(7) = \frac{F'(7)G(7) - F(7)G'(7)}{(G(7))^2} = \frac{\left( \frac{1}{4} \right)(1) - (5)\left( \frac{-2}{3} \right)}{(1)^2}$

Simplifying:

$Q'(7) = \frac{\frac{1}{4} + \frac{10}{3}}{1} = \frac{1}{4} + \frac{10}{3}$

To add these fractions, we find a common denominator:

$Q'(7) = \frac{3}{12} + \frac{40}{12} = \frac{43}{12}$

Thus, the correct value is:

$Q'(7) = \frac{43}{12}$

Recap:

• (a) $P'(2) = \frac{3}{2}$
• (b) $Q'(7) = \frac{43}{12}$

Would you like further details or clarification on any part?

Here are 5 related questions to expand on these concepts:

1. How does the product rule apply when multiplying more than two functions?
2. Can you explain how to estimate derivatives from a graph more precisely?
3. How would $P(x)$ and $Q(x)$ change if one of the functions were a constant?
4. How does the quotient rule differ from the product rule in practical applications?
5. What are common scenarios where using the quotient rule becomes challenging?

Tip: When using the product and quotient rules, carefully check the signs of your derivatives, as small errors here can significantly affect the final result.