A question about Fourier transform of tempered distribution

Question

I have a function $V\in W^{1,1}_\text{loc}(\mathbb{R}^n\times\mathbb{R}^+)$, for $y\in\mathbb{R}^+$, is true that: $V(\cdot,y)\in \mathcal{S}'(\mathbb{R}^n)$, i.e. $V(\cdot,y)$ is a distribution over $\mathbb{R}$? I have no idea on how to prove this statement. Let $\mathcal{F}\colon\mathcal{S}'(\mathbb{R}^n)\to\mathcal{S}'(\mathbb{R}^n)$ the Fourier transform, i don't know how to prove the following two identities: $$ \mathcal{F}(\partial_jV(\cdot,y))=i\xi_j\mathcal{F}(V(\cdot,y)),\quad\forall j=1,...,n,\tag{1}$$ and: $$ \mathcal{F}(\partial_y V(\cdot,y))=\partial_y\mathcal{F}(V(\cdot,y)).\tag{2}$$ If i prove that $V(\cdot,y)\in\mathcal{S}'(\mathbb{R}^n)$, then (1) follow from the well known properties of fourier transform. For (2) i have no idea on how to prove it. Any help would be appreciated.