Working with abstract vectors

You need to tell Mathematica that both $V[t]$ and $V'[t]$ are vectors:

$Assumptions = (V[t]|V'[t]) ∈ Vectors[{n}];

TensorReduce @ D[V[t]/Sqrt[V[t] . V[t]], t] //TeXForm

$\frac{V'(t)}{\sqrt{V(t).V(t)}}-\frac{V(t) V(t).V'(t)}{(V(t).V(t))^{3/2}}$