The derivative of $\sin(\omega_o t)$ is $\cos(\omega_o t)$.
The Fourier transform of $\sin(\omega_o t)$ is $\frac{\pi}{j}[\delta(\omega-\omega_o) - \delta(\omega+\omega_o)]$.
Differentiation in the time domain is equivalent to multiplying the transform by $j\omega$.
The transform of $\cos(\omega_o t)$ is $\pi[\delta(\omega-\omega_o) + \delta(\omega+\omega_o)]$.
What I don't understand is how multiplying the transform of $\sin(\omega_o t)$ by $j\omega$ gives you the transform of $\cos(\omega_o t)$. I see how the $j$'s will cancel out, but how does the sign of that impulse get flipped?