Answer
\[(a) \; F'(2)=20\\
(b) \; G'(3)=63\]
Work Step by Step
\[(a)\; F(x)=f(f(x))\]
Differentiate with respect to $x$ using chain rule
\[F'(x)=f'(f(x))\cdot f'(x)\]
\[F'(2)=f'(f(2))\cdot f'(2)\]
Using data as per the question
\[f(2)=1\;,\; f'(2)=5\]
\[\Rightarrow F'(2)=f'(1)\cdot (5)\]
Using \[f'(1)=4\]
\[F'(2)=(4)(5)=20\]
Hence, $F'(2)=20$.
\[(b) G(x)=g(g(x))\]
Differentiate with respect to $x$ using chain rule
\[G'(x)=g'(g(x))\cdot g'(x)\]
\[G'(3)=g'(g(3))\cdot g'(3)\]
Using given data as per the question \[g(3)=2\;,\; g'(3)=9\]
\[\Rightarrow G'(3)=g'(2)\cdot (9)\]
Using $g'(2)=7$
\[G'(3)=(7)(9)=63\]
Hence, $G'(3)=63$.
