We'll use that:
[tex]g(x)=e^{f(x)}\\\\ g'(x)=f'(x)e^{f(x)}[/tex]
So:
[tex]f(t)=e^{8t\sin(2t)}\\\\ f'(t)=[8t\sin(2t)]'e^{8t\sin(2t)}\\\\ f'(t)=[(8t)'\sin(2t)+8t(\sin(2t))']e^{8t\sin(2t)}\\\\ f'(t)=[8\sin(2t)+8t(2\cos(2t))]e^{8t\sin(2t)}\\\\ \boxed{f'(t)=8(\sin(2t)+2t\cos(2t))e^{8t\sin(2t)}}[/tex]
