If $f$ is a Schwartz function, then $(\widehat{\partial_x f})(\xi)=(-i\xi\widehat{f}(\xi))$ (https://www.amazon.com/-/es/Man-wah-Wong-ebook/dp/B00W8OO57A)
If $f\in L^2(\mathbb{R}^n)$ Does $\widehat{\partial_x f}(\xi)=(-i\xi\widehat{f}(\xi))(x)$ also hold?