Answer
$$\frac{{dy}}{{dx}} = 3\left( {\frac{{2{x^3} + 5x}}{{{x^4} + 5{x^2}}}} \right)$$
Work Step by Step
$$\eqalign{
& y = \ln {\left( {{x^4} + 5{x^2}} \right)^{3/2}} \cr
& {\text{use the power property of logarithms}} \cr
& y = \frac{3}{2}\ln \left( {{x^4} + 5{x^2}} \right) \cr
& {\text{differentiate with respect to }}x \cr
& \frac{{dy}}{{dx}} = \frac{d}{{dx}}\left[ {\frac{3}{2}\ln \left( {{x^4} + 5{x^2}} \right)} \right] \cr
& \frac{{dy}}{{dx}} = \frac{3}{2}\frac{d}{{dx}}\left[ {\ln \left( {{x^4} + 5{x^2}} \right)} \right] \cr
& {\text{use }}\frac{d}{{dx}}\ln g\left( x \right) = \frac{{g'\left( x \right)}}{{g\left( x \right)}} \cr
& \frac{{dy}}{{dx}} = \frac{3}{2}\left( {\frac{{\left( {{x^4} + 5{x^2}} \right)'}}{{{x^4} + 5{x^2}}}} \right) \cr
& \frac{{dy}}{{dx}} = \frac{3}{2}\left( {\frac{{4{x^3} + 10x}}{{{x^4} + 5{x^2}}}} \right) \cr
& \frac{{dy}}{{dx}} = 3\left( {\frac{{2{x^3} + 5x}}{{{x^4} + 5{x^2}}}} \right) \cr} $$