Convolution notation.

Question

If $s(\cdot,\cdot),\frac{\partial s}{\partial x}(\cdot,\cdot) \in C(\mathbb{R}\times\mathbb{R^{+}}), K(\cdot)\in C(\mathbb{R}) $ are integrable. The convolution between $\frac{\partial s}{\partial x}(\cdot,t)$ and $K(\cdot)$ is: $$ (\frac{\partial s}{\partial x}*K)(x,t)=\int_\mathbb{R}{K(\xi)\frac{\partial s}{\partial x}(x-\xi,t)}d\xi . $$ Assuming that the convolution commutes, then $$(K*\frac{\partial s}{\partial x})(x,t)=\int_\mathbb{R}{K(x-\xi)\frac{\partial s}{\partial x}(\xi,t)}d\xi .$$ or $$ (K*\frac{\partial s}{\partial x})(x,t)=\int_\mathbb{R}{K(x-\xi)\frac{\partial s}{\partial \xi}(\xi,t)}d\xi.$$ ??