Check out Szeliski's book:
There is also an old book on feature detection:
You can read the sections that you care about. Also, I think it is always a good idea to read about scale space theory if you are to ...
I think the first smoothing, by $\sigma_D$, is only done to get more stable derivatives whereas in the second step the convolution by a Gaussian with $\sigma_I$ is done to establish the 'scale-space' in which the operator is applied.
Ignoring $\sigma_D$, this looks like this in Matlab:
dx = [-1 0 1; -1 0 1; -1 0 1]; % Simple mask for derivative
Ix = ...