The result if order of two filter are reversed
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.