$f \in W^{1,\infty} \cap L^1 \implies \int_{R^n}f(x,\xi)g(\xi)d\xi \in W^{1,\infty}(\mathbb{R}^n)$

Question

Suppose $g \in L^\infty(\mathbb{R}^m).$ How can I prove that $$f \in W^{1,\infty}(\mathbb{R}^n \times \mathbb{R}^m) \cap L^1(\mathbb{R}^n \times \mathbb{R}^m) \implies \int_{R^n}f(x,\xi)g(\xi)d\xi \in W^{1,\infty}(\mathbb{R}^n)$$