Let $V \subset H \subset V'$ be a Hilbert triple.

We can define a weak derivative of $u \in L^2(0,T;V)$ as the element $u' \in L^2(0,T;V')$ satisfying $$\int_0^T u(t)\varphi'(t)=-\int_0^T u'(t)\varphi(t)$$ for all $\varphi \in C_c^\infty(0,T)$.

Then we define the space $W = \{u \in L^2(0,T;V) : u' \in L^2(0,T;V')\}$. We know for example that for $u, v \in W$, $$\frac{d}{dt}(u(t),v(t))_H = \langle u'(t), v(t) \rangle + \langle v'(t), u(t) \rangle.\tag{1}$$

Now suppose I change my space of test functions and define a weak derivative as an element $u' \in L^2(0,T;V')$ satisfying $$\int_0^T u(t)\varphi'(t)=-\int_0^T u'(t)\varphi(t)$$ for all $\varphi \in C_c^1(0,T)$.

Define $\tilde W = \{u \in L^2(0,T;V) : u' \in L^2(0,T;V')\}$ where the derivative is now with respect to these new test functions. How is this related to $W$? Do properties like (1) still hold?

I think yes, by uniqueness of weak derivatives. But I wanted to check in case I missed something.