# Chapter 4 - Calculating the Derivative - 4.2 Derivatives of Products and Quotients - 4.2 Exercises - Page 216: 9

${p^,}\,\left( y \right) = \, - 8{y^{ - 5}} + 15{y^{ - 6}} + 30{y^{ - 7}}$

#### Work Step by Step

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

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