# A table of values for f,g, f', and g' is

A table of values for f,g, f', and g' is given.
$\begin{array}{|c|c|}\hline x&f(x)&g(x)&f'(x)&g'(x)\\\hline1&3&2&4&6\\\hline2&1&8&5&7\\\hline3&7&2&7&9\\\hline\end{array}$
a) If h(x)=f(g(x)), find h'(3)
b) If H(x)=g(f(x)),find H'(1).

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Lible1953
Given:
To determine:
a) $$\displaystyle{h}{\left({x}\right)}={f{{\left({g{{\left({x}\right)}}}\right)}}}$$
To find h'(3)
$$\displaystyle{h}{\left({x}\right)}={f{{\left({g{{\left({x}\right)}}}\right)}}}$$
Then, $$\displaystyle{h}'{\left({x}\right)}={f}'{\left({g{{\left({x}\right)}}}\right)}\cdot{g}'{\left({x}\right)}$$
Putting x=3,
$$\displaystyle{h}'{\left({3}\right)}={f}'{\left({g{{\left({3}\right)}}}\right)}\cdot{g}'{\left({3}\right)}$$
From the given table,
$$\displaystyle{g{{\left({3}\right)}}}={2}$$ and $$\displaystyle{g}'{\left({3}\right)}={9}$$
So, $$\displaystyle{h}'{\left({3}\right)}={f}'{\left({2}\right)}\cdot{9}$$
From the table, $$\displaystyle{f}'{\left({2}\right)}={5}$$
$$\displaystyle{h}'{\left({3}\right)}={5}\cdot{9}$$
$$\displaystyle{h}'{\left({3}\right)}={45}$$
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Fearen

b) H(x)=g(f(x))
Then,
$$\displaystyle{H}'{\left({x}\right)}={g}'{\left({f{{\left({x}\right)}}}\right)}\cdot{f}'{\left({x}\right)}$$
Putting x=1
$$\displaystyle{H}'{\left({1}\right)}={g}'{\left({f{{\left({1}\right)}}}\right)}\cdot{f}'{\left({1}\right)}$$
From the given table, $$\displaystyle{f{{\left({1}\right)}}}={3}$$ and $$\displaystyle{f}'{\left({1}\right)}={4}$$
$$\displaystyle{H}'{\left({1}\right)}={9}\cdot{4}$$
$$\displaystyle{H}'{\left({1}\right)}={36}$$