Answer
$$ - 29$$
Work Step by Step
$$\eqalign{
& y = \left( {\frac{{3x + 2}}{x}} \right)\left( {{x^{ - 5}} + 1} \right) \cr
& {\text{distribute }} \cr
& y = \left( {\frac{{3x}}{x} + \frac{2}{x}} \right)\left( {{x^{ - 5}} + 1} \right) \cr
& y = \left( {3 + 2{x^{ - 1}}} \right)\left( {{x^{ - 5}} + 1} \right) \cr
& {\text{product rule }} \cr
& \left( {uv} \right)' = uv' + vu' \cr
& = \left( {3 + 2{x^{ - 1}}} \right)\left( {{x^{ - 5}} + 1} \right)' + \left( {{x^{ - 5}} + 1} \right)\left( {3 + 2{x^{ - 1}}} \right)' \cr
& = \left( {3 + 2{x^{ - 1}}} \right)\left( { - 5{x^{ - 6}}} \right) + \left( {{x^{ - 5}} + 1} \right)\left( { - 2{x^{ - 2}}} \right) \cr
& {\text{evaluate }}{\left. {dy/dx} \right|_{x = 1}} \cr
& = \left( {3 + 2{{\left( 1 \right)}^{ - 1}}} \right)\left( { - 5{{\left( 1 \right)}^{ - 6}}} \right) + \left( {{{\left( 1 \right)}^{ - 5}} + 1} \right)\left( { - 2{{\left( 1 \right)}^{ - 2}}} \right) \cr
& {\text{simplify}} \cr
& = \left( {3 + 2} \right)\left( { - 5} \right) + \left( {1 + 1} \right)\left( { - 2} \right) \cr
& = - 29 \cr} $$
