10. The smallest positive period of the function $y=\sin x-2 \cos 2 x+4 \sin 4 x$ is $\qquad$
10. $2 \pi$. Because the smallest positive period of $\sin x$ is $T_{1}=2 \pi$, the smallest positive period of $\cos 2 x$ is $T_{2}=\pi$, and the smallest positive period of $\sin 4 x$ is $T_{3}=\frac{\pi}{2}$. The least common multiple of $2 \pi, \pi, \frac{\pi}{2}$ is $2 \pi$. Therefore, the smallest positive period of the function $y=\sin x-2 \cos 2 x+4 \sin 4 x$ is $2 \pi$.