Can a curve's unit normal vector be determined using the second derivative? Because $$T = \frac {r'}{|r'|}$$ I was wondering whether or not it was also valid to solve for the unit normal vector with the second derivative without first solving for T: $$N = \frac {r''}{|r''|}$$

There is a theorem that says that if a vector function has constant length then its derivative vector will be orthogonal to it. Since T has constant length (because it is a unit vector) then T' is guaranteed to be orthogonal to T and hence normal to the curve. 
On the other hand r' does not necessarily have constant length and hence r'' doesnt have to be orthogonal to r' and so not normal to the curve.
