find T(t), N(t), $\displaystyle a_t , a_n $ at the given time t for the plane curve r(t)

$\displaystyle r(t)= ti + \frac{1}{t} j, t = 1$

The book gives this formula for T(t)

$\displaystyle T(t) = \frac {r'(t)}{ || r'(t) ||}$

So, i find $\displaystyle r' = i - \frac {1}{t^2} j$

then the magnitude

$\displaystyle ||r'(t)|| = \sqrt{ (\frac{1^2 + 1^2}{t^2})^2} = \sqrt{ \frac{2}{t^4}} $

then, combining for T(t) the solutions manual has this which i cant seem to understand.

$\displaystyle T(t) = \frac {t^2} {\sqrt{ t^4 + 1}} ( i - \frac{1} {\sqrt{ {t^2}}}j) = \frac{1} {\sqrt{ t^4 + 1}}(t^2i - j)$

I am completely lost!!!