1st Order Taylor Expansion

I need to do a first order expansion of g(k + q) about q - where q is small and k is fixed (k may be large). I would actually call it more of a first order approximation. The function g depends of two variables and is given by

g(k) = B^2 + 4[t + 2tcos(1/2(sqrt(3))ak_x)cos((ak_y)/2)]^2
- Bsqrt[B^2 + 12t^2 + 8t^2(2cos(1/2(sqrt(3))ak_x)cos((ak_y)/2) + cos(ak_y)]

where k=(k_x,k_y).

See attached PDF.

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Solution Summary

We find the first order approximation, or first order Taylor expansion, of a function of 2 variables