Answer
\[{p^,}\,\left( y \right) = \, - 8{y^{ - 5}} + 15{y^{ - 6}} + 30{y^{ - 7}}\]
Work Step by Step
\[\begin{gathered}
p\,\left( y \right) = \,\left( {{y^{ - 1}} + {y^{ - 2}}} \right)\,\left( {2{y^{ - 3}} - 5{y^{ - 4}}} \right) \hfill \\
Use\,\,the\,\,product\,\,rule\,\,to\,\,find\,\,{p^,}\,\left( y \right) \hfill \\
{p^,}\,\left( y \right) = \,\left( {{y^{ - 1}} + {y^{ - 2}}} \right)\,{\left( {2{y^{ - 3}} - 5{y^{ - 4}}} \right)^,} + \,\left( {2{y^{ - 3}} - 5{y^{ - 4}}} \right)\,{\left( {{y^{ - 1}} + {y^{ - 2}}} \right)^,} \hfill \\
Then \hfill \\
{p^,}\,\left( y \right) = \,\left( {{y^{ - 1}} + {y^{ - 2}}} \right)\,\left( { - 6{y^{ - 4}} + 20{y^{ - 5}}} \right) + \,\left( {2{y^{ - 3}} - 5{y^{ - 4}}} \right)\,\left( { - {y^{ - 1}} - 2{y^{ - 3}}} \right) \hfill \\
Simplify\,\,by\,\,multiplying\,\,and\,\,combining\,\,terms \hfill \\
{p^,}\,\left( y \right) = \, - 6{y^{ - 5}} + 20{y^{ - 6}} - 6{y^{ - 6}} + 20{y^{ - 7}} - 2{y^{ - 5}} - 4{y^{ - 6}} + 5{y^{ - 6}} + 10{y^{ - 7}} \hfill \\
{p^,}\,\left( y \right) = \, - 8{y^{ - 5}} + 15{y^{ - 6}} + 30{y^{ - 7}} \hfill \\
\end{gathered} \]
