Single Variable Calculus, Chapter 3, Review Exercises, Section Review Exercises, Problem 60

Suppose that $f$ and $g$ are the functions whose graphs are shown, let $P(x) = f(x) g(x), \displaystyle Q(x) = \frac{f(x)}{g(x)}$ and $C(x) = f(g(x))$. Find a.) $P'(2)$, b.) $Q'(2)$ and c.) $C'(2)$.

*Refer to the graph in the book.

a.) $P'(2)$


$
\begin{equation}
\begin{aligned}

P'(x) =& \frac{d}{dx} f(x) g(x)
\\
\\
P'(x) =& f(x) \frac{d}{dx} g(x) + g(x) \frac{d}{dx} f(x)
\\
\\
P'(x) =& f(x) g'(x) + g(x) f'(x)
\\
\\
P'(2) =& f(2) g'(2) + g(2) f'(2)
\\
\\
P'(2) =& (1) \left( \frac{4}{2} \right) + (4)\left( \frac{2}{-2} \right)
\\
\\
P'(2) =& 2 + (4)(-1)
\\
\\
P'(2) =& 2 - 4
\\
\\
P'(2) =& -2

\end{aligned}
\end{equation}
$


b.) $Q'(2)$


$
\begin{equation}
\begin{aligned}

Q'(x) =& \frac{d}{dx} \frac{f(x)}{g(x)}
\\
\\
Q'(x) =& \frac{\displaystyle g(x) \frac{d}{dx} f(x) - f(x) \frac{d}{dx} g(x)}{[g(x)]^2}
\\
\\
Q'(x) =& \frac{g(x) f'(x) - f(x) g'(x)}{g^2(x)}
\\
\\
Q'(2) =& \frac{g(2) f'(2) - f(2) g'(2)}{g^2 (2)}
\\
\\
Q'(2) =& \frac{(4) \left( \frac{-2}{2} \right) - (1) \left( \frac{4}{2} \right)}{(4)^2}
\\
\\
Q'(2) =& \frac{(4)(-1) - (1)(2)}{16}
\\
\\
Q'(2) =& \frac{-4 - 2}{16}
\\
\\
Q'(2) =& \frac{-6}{16}
\\
\\
Q'(2) =& \frac{-3}{8}

\end{aligned}
\end{equation}
$