x′ = r′×cos(θ) - r×sin(θ)×θ′
y′ = r′×sin(θ) + r×cos(θ)×θ′
That's fine, but we usually want the velocity expressed in the directions of r and θ, rather than the x and y coordinates. To switch to the new orthonormal basis, take the dot product of the above velocity vector with the unit radial vector and the unit tangent vector. These vectors are cos(θ),sin(θ) and -sin(θ),cos(θ) respectively. After taking dot products, the velocity, measured along r and θ, is r′ and r×θ′. The radial speed is the change in r, and the tangential speed is the change in θ times r. When the particle is far from the origin, a small change in θ makes a big difference.
Differentiate again, using the chain rule, to get acceleration. Dot this with the radial and tangential unit vectors, as we did with velocity. I'll spare you the algebra. The acceleration away from the origin is r′′-r×θ′2. The first term is the radial acceleration, and the second is the centripital acceleration. If the partical is tracing a perfect circle, the second term gives the acceleration needed to keep the particle in its orbit.
The acceleration in the tangential direction is rθ′′+2r′θ′. The first term is angular acceleration magnified by the radial distance, and the second term is the coriolis force. It takes extra force to keep a skater spinning at the same rate, as she extends her arms. Her hands, moving outward, represent r′×θ′.