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Section 3-4 : Product and Quotient Rule

7. If\(f\left( 2 \right) = - 8\), \(f'\left( 2 \right) = 3\), \(g\left( 2 \right) = 17\) and \(g'\left( 2 \right) = - 4\) determine the value of \({\left( {f\,g} \right)^\prime }\left( 2 \right)\).

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We know that the product rule is,

\[{\left( {f\,g} \right)^\prime }\left( x \right) = f'\left( x \right)g\left( x \right) + f\left( x \right)g'\left( x \right)\]

Now, we want to know the value of this at \(x = 2\) and so all we need to do is plug this into the derivative. Doing this gives,

\[{\left( {f\,g} \right)^\prime }\left( 2 \right) = f'\left( 2 \right)g\left( 2 \right) + f\left( 2 \right)g'\left( 2 \right)\]

Now, we were given values for all these quantities and so all we need to do is plug these into our “formula” above.

\[{\left( {f\,g} \right)^\prime }\left( 2 \right) = \left( 3 \right)\left( {17} \right) + \left( { - 8} \right)\left( { - 4} \right) = \require{bbox} \bbox[2pt,border:1px solid black]{{83}}\]