Answer
\[{q^,}\,\left( x \right) = - 24{x^{ - 7}} + 28{x^{ - 8}} - 9{x^{ - 4}} + 12{x^{ - 5}}\]
Work Step by Step
\[\begin{gathered}
q\,\left( x \right) = \,\left( {{x^{ - 2}} - {x^{ - 3}}} \right)\,\left( {3{x^{ - 1}} + 4{x^{ - 4}}} \right) \hfill \\
Use\,\,the\,\,product\,\,rule\,\,to\,\,find\,\,q{\,^,}\left( x \right) \hfill \\
q{\,^,}\left( x \right) = \,\left( {{x^{ - 2}} - {x^{ - 3}}} \right)\,{\left( {3{x^{ - 1}} + 4{x^{ - 4}}} \right)^,} + \,\left( {3{x^{ - 1}} + 4{x^{ - 4}}} \right)\,\left( {{x^{ - 2}} - {x^{ - 3}}} \right) \hfill \\
Then \hfill \\
{q^,}\,\left( x \right) = \left( {{x^{ - 2}} - {x^{ - 3}}} \right)\,\,\left( { - 3{x^{ - 2}} - 16{x^{ - 5}}} \right) + \,\left( {3{x^{ - 1}} + 4{x^{ - 4}}} \right)\,\left( { - 2{x^{ - 3}} + 3{x^{ - 4}}} \right) \hfill \\
Simplify\,\,by\,\,multiplying\,\,and\,\,combining\,\,terms \hfill \\
{q^,}\,\left( x \right) = - 3{x^{ - 4}} - 16{x^{ - 7}} + 3{x^{ - 5}} + 16{x^{ - 8}} - 6{x^{ - 4}} + 9{x^{ - 5}} - 8{x^{ - 7}} + 12{x^{ - 8}} \hfill \\
{q^,}\,\left( x \right) = - 24{x^{ - 7}} + 28{x^{ - 8}} - 9{x^{ - 4}} + 12{x^{ - 5}} \hfill \\
\hfill \\
\end{gathered} \]
