Given a test function $\varphi \in C_0^{\infty}$ we have that $d\varphi (x,v) = D\varphi(x)\cdot v$.
I read that one has to use the chain rule to prove that but I'm not sure how to incorporate it here?
These are the definitions:
$$df(x,v) := lim_{t \rightarrow 0} \frac{f(x+tv)-f(x)}{t}$$ $$Df(x)=(D_1f(x),\dots,D_nf(x))$$ with $D_{i}f(x)=\frac{\partial f}{\partial x_{i}}(x)$ and $C_0^{\infty}$ is the space of all test functions.