Paul's Online Notes
Paul's Online Notes
Home / Calculus I / Derivatives / Product and Quotient Rule
Show Mobile Notice Show All Notes Hide All Notes
Mobile Notice
You appear to be on a device with a "narrow" screen width (i.e. you are probably on a mobile phone). Due to the nature of the mathematics on this site it is best views in landscape mode. If your device is not in landscape mode many of the equations will run off the side of your device (should be able to scroll to see them) and some of the menu items will be cut off due to the narrow screen width.

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 \({\left( {f\,g} \right)^\prime }\left( 2 \right)\).

Show Solution

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.

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