Differentiation commutes with convolution ($D^{\alpha }(T\ast \varphi )=(D^{\alpha }T)\ast \varphi =T\ast (D^{\alpha }\varphi )$)

For a distribution and a compactly supported smooth kernel, differentiation with constant coefficients commutes with convolution wherever the kernel support stays inside the domain. The identity follows by testing against the translated kernel and tracking the sign of its derivative. A variable coefficient must remain inside the product before convolution; in general $(bu{)}_{\sigma }\ne b{u}_{\sigma }$.