Suppose $\phi:H^1(S) \to H^2(R)$ is a linear homeomorphism, where $S$ and $R$ are compact surfaces in $\mathbb{R}^n$. Let $\varphi \in D(0,T;H^1(S))$ be a $H^1(S)$-valued $C_c^\infty(0,T)$ function. So $\varphi(t) \in H^1(S)$ for each $t$.
Recall that a weak derivative of a function $u$ is a function $u'$ that satisfies $\int_0^T uv' = -\int_0^T u'v$ for all $v \in C_c^\infty(0,T)$.
I want to show that the weak derivative of the homeomorphism of $\varphi$ is the homeomorphism of the weak derivative of $\varphi$, i.e. that $\phi(\varphi'(t)) = [\phi(\varphi(t))]'$
How do I show this rigorously? I can sort of see that since $\phi$ just acts in space, the prime on the RHS goes straight through to the argument of $\phi$, but that is not a proper argument.