Finding the unit tagent vector, normal vec and curvature problem

[mr_coffee]

Hello everyone, the problem says to:
For the curve gien by r(t) = <1/3* t^3, 1/2 * t^2, t>
find (a) The unit tagent vector;
(b) the unit normal vector;
(c) the curvature;

Well it seems easy enough! the formula's are just derivatives for instance:
The unit tagent vector says:
T(t) = r'(t)/|r'(t)| i got this one right, you can see my work on the image below:
but part (b) i missed..
The normal vector N is suppose to just be:
N(t) = T'(t)/|T'(t)|;
Here is my work and it does not match the back of the book.
http://show.imagehosting.us/show/800636/0/nouser_800/T0_-1_800636.jpg

 

[HallsofIvy]

In finding N, you did NOT find T'/|T'|. You used r' rather than T.

Yes, $$T= <t^2, t, 1>/\sqrt{t^4+ t^2+ 1}$$.
To find N(t) you have to differentiate THAT: differentiate
$$\left<\frac{t^2}{\sqrt{t^4+ t^2+ 1}},\frac{t}{\sqrt{t^4+ t^2+ 1}},\frac{1}{\sqrt{t^4+ t^2+ 1}}\right>$$.