Finding Filtered Output using Convolution property of Continuous Fourier Transform
\$\delta (t)\$ is the unit impulse and is zero everywhere except at \$t=0\$, and \$sin\:(\omega _0 t)=0\$ at \$t=0\$. The product of these two is zero everywhere.

\$u(t)\$ is the unit step function and its derivative is infinite at \$t=0\$ and zero everywhere else, and this is the definition of the unit impulse.