Answer
$$\frac{{dy}}{{dx}} = - 12\cosh 4x{\sinh ^2}4x$$
Work Step by Step
$$\eqalign{
& y = - {\sinh ^3}4x \cr
& y = - {\left( {\sinh 4x} \right)^3} \cr
& {\text{computing }}dy/dx \cr
& \frac{{dy}}{{dx}} = - \frac{d}{{dx}}{\left( {\sinh 4x} \right)^3} \cr
& {\text{by the chain rule}} \cr
& \frac{{dy}}{{dx}} = - 3{\left( {\sinh 4x} \right)^2}\frac{d}{{dx}}\left( {\sinh 4x} \right) \cr
& \frac{{dy}}{{dx}} = - 3{\left( {\sinh 4x} \right)^2}\left( {\cosh 4x} \right)\frac{d}{{dx}}\left( {4x} \right) \cr
& \frac{{dy}}{{dx}} = - 3{\left( {\sinh 4x} \right)^2}\left( {\cosh 4x} \right)\left( 4 \right) \cr
& {\text{multiplying}} \cr
& \frac{{dy}}{{dx}} = - 12\cosh 4x{\sinh ^2}4x \cr} $$
