Answer
$J' = - \left( {\frac{3}{{{u^4}}} + \frac{2}{{{u^3}}} + \frac{1}{{{u^2}}}} \right)$
Work Step by Step
$$\eqalign{
& J = \left( {\frac{1}{u} + \frac{1}{{{u^2}}}} \right)\left( {u + \frac{1}{u}} \right) \cr
& {\text{Rewrite the function using the property }}\frac{1}{{{x^n}}} = {x^{ - n}} \cr
& J = \left( {{u^{ - 1}} + {u^{ - 2}}} \right)\left( {u + {u^{ - 1}}} \right) \cr
& {\text{Differentiating}} \cr
& J' = \frac{d}{{du}}\left[ {\left( {{u^{ - 1}} + {u^{ - 2}}} \right)\left( {u + {u^{ - 1}}} \right)} \right] \cr
& {\text{Differentiate by using the formula }}\left( {fh} \right)' = fh' + hf' \cr
& {\text{Let }}f = {u^{ - 1}} + {u^{ - 2}}{\text{ and }}h = u + {u^{ - 1}},{\text{ then}} \cr
& J' = \left( {{u^{ - 1}} + {u^{ - 2}}} \right)\left( {u + {u^{ - 1}}} \right)' + \left( {u + {u^{ - 1}}} \right)\left( {{u^{ - 1}} + {u^{ - 2}}} \right)' \cr
& {\text{Compute derivatives}} \cr
& J' = \left( {{u^{ - 1}} + {u^{ - 2}}} \right)\left( {1 - {u^{ - 2}}} \right) + \left( {u + {u^{ - 1}}} \right)\left( { - {u^{ - 2}} - 2{u^{ - 3}}} \right) \cr
& {\text{Multiply and simplify}} \cr
& J' = {u^{ - 1}} - {u^{ - 3}} + {u^{ - 2}} - {u^{ - 4}} - {u^{ - 1}} - 2{u^{ - 2}} - {u^{ - 3}} - 2{u^{ - 4}} \cr
& J' = - 3{u^{ - 4}} - 2{u^{ - 3}} - {u^{ - 2}} \cr
& {\text{Rewrite the function using the property }}\frac{1}{{{x^n}}} = {x^{ - n}} \cr
& J' = - \frac{3}{{{u^4}}} - \frac{2}{{{u^3}}} - \frac{1}{{{u^2}}} \cr
& J' = - \left( {\frac{3}{{{u^4}}} + \frac{2}{{{u^3}}} + \frac{1}{{{u^2}}}} \right) \cr} $$
