Polar/velocity problem

(a.) A particle moving with a nonzero velocity along the polar curve given by r = 3 + 2 costheta has position (x(t), y(t)) at theta = 0 when t = 0. This particle moves along the curve so that dr/dt = dr/dtheta. Find the value of dr/dt at theta = pi/3 and interpret your answer in terms of the motion of the particle.

(b.) For the particle described above, dy/dt = dy/dtheta. Find the value of dy/dt at theta = pi/3. Interpret your answer in terms of the motion of the particle.

I found the above problems to be kinda confusing. For the first part I answered "0.115 rads/time counterclockwise" and for the second part, "0.5 rads/time vertically." I've no idea if I did the problem right. I think I'm confused because the slope of polar curves are supposed to be the same thing as rate but what about when the slope is exactly vertical? Is the velocity infinitely fast then? But the particle is actually barely moving away from the origin of the polar graphs.... (unlike standard equations)

I'd appreciate some clarification very much...

Another question... does dr/dtheta actually exist? I learned that it really doesn't exist; you have to use dy/dx. Is this true? Also, are rate and slope still the same thing in polar graphs?