The result if order of two filter are reversed Apply the Composite Laplacian Filter first, then apply the gaussian filter.
Apply the gaussian filter first, then apply the composite laplacian filter.

My work as below:
Here we assume the original image is function $f(x,y)$
$$ (1): (f-\nabla^2f)\ast G = f \ast G - \nabla^2f \ast G$$
$$ (2): (f \ast G) - \nabla^2(f \ast G) = f \ast G - f \ast \nabla^2G $$
I am wondering the statement $\nabla^2f \ast G = f \ast \nabla^2G$ is true or not.
Formally, (linear) derivatives and convolutions may commute, as explained on Wikipedia-Convolution/Properties/Differentiation. This is a major &quot;operation-saving&quot; property: if one wants to differentiate many images and convolve then with a fixed kernel (here a Gaussian), one can convolve the images with the differentiate kernel.
You don't have to differentiate all images, just the kernel once.
Caveat: what is written above could be taken with care, notably when derivatives don't exist theoretically, or with subtleties in implementing derivatives.