Thank you. I'd tried the process of working backward by differentiating the textbook answer, but completely forgot the need to use the chain rule when doing so. I really appreciate the help.
Homework Statement
The question asks one to derive the acceleration vector,
$$\vec{a} = [\frac{d^2r}{dt^2} - r(\frac{d\Theta}{dt})^2]\vec{u}_r + [\frac{1}{r}\frac{d}{dt}(r^2\frac{d\Theta }{dt})]\vec{u}_\Theta$$
from the velocity vector.
$$\vec{v} = \vec{u}_r\frac{dr}{dt} +...