Think of it like the Taylor expansion of the variance. What you have above is the second order expansion. Note that you get some nice cancelations in your taylor series.
See: page B-5 of
Just got out of my Stats midterm and one of the questions was: Find the approximate variance of your MEM estimator.
I haven't used Delta in a while, and don't think I would have thought of it, if I hadn't been floating it around in my head the night before.
Thanks. You are awesome for asking this question.