Answer
$$\frac{{dy}}{{dx}} = - \frac{{15\sin x}}{{{{\left( {5 - \cos x} \right)}^2}}}$$
Work Step by Step
$$\eqalign{
& y = \frac{{3\cos x}}{{5 - \cos x}} \cr
& {\text{differentiate with respect to }}x \cr
& \frac{{dy}}{{dx}} = {D_x}\left[ {\frac{{3\cos x}}{{5 - \cos x}}} \right] \cr
& {\text{use the quotient rule for derivatives}} \cr
& \frac{{dy}}{{dx}} = \frac{{\left( {5 - \cos x} \right) \cdot {D_x}\left( {3\cos x} \right) - 3\cos x \cdot {D_x}\left( {5 - \cos x} \right)}}{{{{\left( {5 - \cos x} \right)}^2}}} \cr
& {\text{solve derivatives }}{D_x}\left( {\cos x} \right) = - \sin x \cr
& \frac{{dy}}{{dx}} = \frac{{\left( {5 - \cos x} \right)\left( { - 3\sin x} \right) - 3\cos x\left( {\sin x} \right)}}{{{{\left( {5 - \cos x} \right)}^2}}} \cr
& \frac{{dy}}{{dx}} = \frac{{ - 15\sin x + 3\cos x\sin x - 3\cos x\sin x}}{{{{\left( {5 - \cos x} \right)}^2}}} \cr
& {\text{simplifying}} \cr
& \frac{{dy}}{{dx}} = - \frac{{15\sin x}}{{{{\left( {5 - \cos x} \right)}^2}}} \cr} $$