Laplacian of Gaussian operator
This sounds counter intuitive to many, but as long as the difference operator and the smoothing kernel are linear and space-invariant, they can be applied in any order, and thus are often combined in a single convolution operator (for more computational efficiency), for the same.
For some intuition, consider that, either for the linear smoothing and the derivative, a noisy pixel  is replaced by a linear weighed combination of other pixels. And a linearly-weighed combination of another linearly-weighed combination remains a linear combination, the same due to the commutativity and the associativity properties of the addition and the multiplication.
This is illustrated below for the first derivative only, but you can follow the same argument by differentiating again.


What you describe is commonly known as the LoG, Laplacian of Gaussian.
Note that when one operation becomes nonlinear, this is not true anymore.