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

skeexerxo175o 2021-11-21 Answered
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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Expert Answer

Lible1953
Answered 2021-11-22 Author has 1818 answers
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
Answered 2021-11-23 Author has 1068 answers

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 \(\displaystyle{f}'{\left({1}\right)}={4}\)
\(\displaystyle{H}'{\left({1}\right)}={9}\cdot{4}\)
\(\displaystyle{H}'{\left({1}\right)}={36}\)

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