Dot product of f ' (t) and f (t)?

Using the product rule 2f ′(t) · f(t)=(f (t) · f(t))' = [(cos^2 + sin^2 + a^2t^2)/(1+a^2t^2)]' = [(1+a^2t^2)/(1+a^2t^2)]' = 1' = 0.