Answer
\[(a) h'(1)=30
\\
\\
(b) H'(1)=36\]
Work Step by Step
$(a)\;\; h(x)=f(g(x))$ ___(1)
Differentiating (1) with respect to $x$ using chain rule
$h'(x)=f'(g(x))\cdot g'(x)$
$h'(1)=f'(g(1))\cdot g'(1)$
[Using data from given table $g(1)=2\;,\;g'(1)=6$]
$h'(1)=f'(2)\cdot 6$
[Using data from given table $f'(2)=5$]
$h'(1)=5\times 6=30$
Hence $h'(1)=30$.
$(b)\;\; H(x)=g(f(x))$ _____(2)
Differentiating (2) with respect to $x$ using chain rule
$H'(x)=g'(f(x))\cdot f'(x)$
$H'(1)=g'(f(1))\cdot f'(1)$
[Using data from given table $\;f(1)=3\;,\;f'(1)=4$]
$H'(1)=g'(3)\cdot 4$
[Using data from given table $\;g'(3)=9$]
$H'(1)=9\times 4=36$
Hence $H'(1)=36$.
