Single Variable Calculus, Chapter 3, 3.5, Section 3.5, Problem 64

The table of values for $f, g, f'$, and $g'$ is given.

$
\begin{array}{|c|c|c|c|c|}
\hline\\
x & f(x) & g(x) & f'(x) & g'(x) \\
\hline\\
1 & 3 & 2 & 4 & 6 \\
\hline\\
2 & 1 & 8 & 5 & 7 \\
\hline\\
3 & 7 & 2 & 7 & 9\\
\hline
\end{array}
$


a.) Suppose that $F(x) = f(f(x))$, find $F'(2)$.


$
\begin{equation}
\begin{aligned}

F'(x) =& f'(f(x)) f'(x)
\\
\\
F'(2) =& f'(f(2)) f'(2)
\\
\\
F'(2) =& f'(1) \cdot 5
\\
\\
F'(2) =& (4)(5)
\\
\\
F'(2) =& 20

\end{aligned}
\end{equation}
$



b.) Suppose that $G(x) = g(g(x))$, find $G'(3)$.


$
\begin{equation}
\begin{aligned}

G'(x)=& g'(g(x)) g'(x)
\\
\\
G'(3)=& g'(g(3)) g'(3)
\\
\\
G'(3)=& g'(2) \cdot 9
\\
\\
G'(3)=& (7)(9)
\\
\\
G'(3)=& 63


\end{aligned}
\end{equation}
$