Single Variable Calculus, Chapter 3, 3.3, Section 3.3, Problem 68

Let $P(x) = F(x) G(x)$ and $\displaystyle Q(x) = \frac{F(x)}{G(x)}$, where $F$ and $G$ are the functions whose are shown
a.) Find $P'(2) \qquad$ b.) Find $Q'(7)$




a.) $P'(2) = F'(2) [G(2)] + F(2) [G'(2)]$
Referring to the given graph

$\begin{equation} \begin{aligned} F(2)&= 3, \quad F'(2) = 0, \quad G(2) = 2, \quad G'(2) = \frac{1}{2}\\ \\ P'(2)&= 0 (2) + 3\left( \frac{1}{2}\right)\\ \\ P'(2) &= \frac{3}{2} \end{aligned} \end{equation}$


b.) $\displaystyle Q'(7) = \frac{G(7)[F'(7)]-[F(7)]G'(7)}{[G(7)]^2}$
Referring to the graph given,

$\begin{equation} \begin{aligned} F(7) &= 5, \quad F'(7) = \frac{1}{4}, \quad G(7) = 1, \quad G'(7) = \frac{-2}{3}\\ \\ Q'(7) &= \frac{1 \left( \frac{1}{4} \right) - 5 \left( \frac{-2}{3}\right)}{(1)^2}\\ \\ Q'(7) &= \frac{43}{12} \end{aligned} \end{equation}$