Answer
$$\frac{{dy}}{{dx}} = 4{x^3} - \frac{{4{x^2} - 2x - 480}}{{{{\left( {4x - 1} \right)}^2}}}$$
Work Step by Step
$$\eqalign{
& y = {x^4} - \frac{{{x^2} + 120}}{{4x - 1}} \cr
& {\text{Differentiate}} \cr
& \frac{{dy}}{{dx}} = \frac{d}{{dx}}\left[ {{x^4} - \frac{{{x^2} + 120}}{{4x - 1}}} \right] \cr
& {\text{Use sum rule}} \cr
& \frac{{dy}}{{dx}} = \frac{d}{{dx}}\left[ {{x^4}} \right] - \frac{d}{{dx}}\left[ {\frac{{{x^2} + 120}}{{4x - 1}}} \right] \cr
& {\text{Use power rule and quotient rule}} \cr
& \frac{{dy}}{{dx}} = 4{x^3} - \frac{{\left( {4x - 1} \right)\left( {2x} \right) - \left( {{x^2} + 120} \right)\left( 4 \right)}}{{{{\left( {4x - 1} \right)}^2}}} \cr
& {\text{Simplifying}} \cr
& \frac{{dy}}{{dx}} = 4{x^3} - \frac{{8{x^2} - 2x - 4{x^2} - 480}}{{{{\left( {4x - 1} \right)}^2}}} \cr
& \frac{{dy}}{{dx}} = 4{x^3} - \frac{{4{x^2} - 2x - 480}}{{{{\left( {4x - 1} \right)}^2}}} \cr} $$
