Answer
$$\left[ { - 4\pi \cos \left( {4\pi x} \right) - 5\sin \left( {4\pi x} \right)} \right]{e^{5x}}$$
Work Step by Step
$$\eqalign{
& \frac{d}{{dx}}\left[ {{e^{5x}}\sin \left( { - 4\pi x} \right)} \right] \cr
& {\text{Recall that }}\sin \left( { - \theta } \right) = - \sin \theta \cr
& \frac{d}{{dx}}\left[ { - {e^{5x}}\sin \left( {4\pi x} \right)} \right] \cr
& {\text{Differentiate by using the product rule}} \cr
& = - {e^{5x}}\frac{d}{{dx}}\left[ {\sin \left( {4\pi x} \right)} \right] - \sin \left( {4\pi x} \right)\frac{d}{{dx}}\left[ {{e^{5x}}} \right] \cr
& {\text{Computing derivatives}} \cr
& = - {e^{5x}}\cos \left( {4\pi x} \right)\left( {4\pi } \right) - \sin \left( {4\pi x} \right)\left( {5{e^{5x}}} \right) \cr
& {\text{Simplifying}} \cr
& = \left[ { - 4\pi \cos \left( {4\pi x} \right) - 5\sin \left( {4\pi x} \right)} \right]{e^{5x}} \cr} $$
