r(t) = t i + t^2 j
First we find the unit tangent vector
T(t) = (i+2tj)/ sqrt(1+4t^2)
Now use the quotient rule to find T '(t)
T'(t) = (1+4t^2)^1/2 (2j) - (i+2tj) 4t (1+4t^2)^ -1/2
Unit tangent and unit normal vectors are investigated. The solution is detailed and well presented.