Sobolev estimates for products

Question

Let $s>1/2$ fixed, and consider two functions $f\in H^s(\mathbb{R})$ and $g\in W^{s,\infty}(\mathbb{R})$. My question is the following: is it true that there exists a constant $c>0$ such that $$ \Vert fg\Vert_{H^s}\leq c\Vert f\Vert_{H^s}\Vert g\Vert_{W^{s,\infty}}? $$ My main problem with the above inequality is the fact that $s$ is not necessarily natural, so I guess I would need some sort of fractional Leibnitz rule, which I am not familiar with (I know there exists something with that name, but I don't really know about it). It is very easy to see that, as soon as $s\in\mathbb{N}$ then the inequality holds, but what happen when $s\notin\mathbb{N}$?