I'm confused about weak derivative definition.

$u \in L^2(0,T;V)$ has weak derivative $u'\in L^2(0,T;V')$ iff

$$\int_0^T u(t)\varphi'(t) = -\int_0^T u'(t)\varphi(t)$$

holds for all $\varphi \in C_0^\infty(0,T).$

The LHS is an object in $V$. What is the RHS? Should the integrand be $\langle u'(t), \varphi(t) \rangle_{V',V}$ (**edit**: I guess not)? If so then it is a real number. If not, it is in $V'$.

Help would be appreciated