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Section 3-4 : Product and Quotient Rule

8. If \(f\left( x \right) = {x^3}g\left( x \right)\), \(g\left( { - 7} \right) = 2\), \(g'\left( { - 7} \right) = - 9\) determine the value of \(f'\left( { - 7} \right)\).

Hint : Even though we don’t know what \(g\left( x \right)\) is we can still use the product rule to take the derivative and then we can use the given information to get the value of \(f'\left( { - 7} \right)\).
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Even though we don’t know what \(g\left( x \right)\) is we do have a product of two functions here and so we can use the product rule to determine the derivative of \(f\left( x \right)\).

\[f'\left( x \right) = 3{x^2}g\left( x \right) + {x^3}g'\left( x \right)\] Now all we need to do is plug \(x = - 7\) into this and use the given information to determine the value of \(f'\left( { - 7} \right)\).

\[f'\left( { - 7} \right) = 3{\left( { - 7} \right)^2}g\left( { - 7} \right) + {\left( { - 7} \right)^3}g'\left( { - 7} \right) = 3\left( {49} \right)\left( 2 \right) + \left( { - 343} \right)\left( { - 9} \right) = \require{bbox} \bbox[2pt,border:1px solid black]{{3381}}\]