Let $U\subset\mathbb{R}^n$ be an open set and $f\in W^{1,p}(U)$. I wish to prove that $$ \mathcal{F}\left[(D_{x_k}f)(x)\right] = i \xi_k\, \mathcal{F}\left[f(x)\right] $$ where $D_{x_k}f$ is the weak derivative of $f$ with respect to $x_k$.
My naive attempt is to approach it directly: $$ \mathcal{F}\left[(D_{x_k}f)(x)\right] = \int_U\left(D_{x_k}f(x)\right)e^{-i\xi\cdot x}\,dx $$ Since $e^{-i\xi\cdot x}\notin C^{\infty}_c(U)$ so we can't use the definiton of weak derivative directly. Also, since we are dealing with weak derivative instead of usual partial derivative, I am not sure how if we can use integration by parts in the usual sense.
How should I proceed with this? Or is there a better approach than to deal with this integration head-on?