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Suppose I define

F1[R + a] = F1[R] + a D[F1[R],R]
F2[R + 2 a] = F2[R] + 2 a D[F2[R],R]

and I want to expand the square of their sum keeping only first order terms in a.

Expand[(F1[R + a] + F2[R + 2 a])^2]
(*=>F1[R]^2 + 2 F1[R] F2[R] + F2[R]^2 + 2 a F1[R] Derivative[1][F1][R] + 
 2 a F2[R] Derivative[1][F1][R] + a^2 Derivative[1][F1][R]^2 + 
 4 a F1[R] Derivative[1][F2][R] + 4 a F2[R] Derivative[1][F2][R] + 
 4 a^2 Derivative[1][F1][R] Derivative[1][F2][R] + 
 4 a^2 Derivative[1][F2][R]^2 *)

Expand doesn't work with Assumptions. How can I expand the above sum and exclude a^2 terms?

I can do Expand[(F1[R + a] + F2[R + 2 a])^2] /. a^2 -> 0 but I don't really like it.

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Try Series assuming a to be small:

Normal[Series[(F1[R] + a D[F1[R], R] + F2[R] + 2 a D[F2[R],R])^2, {a,0, 1}]]
(*(F1[R] + F2[R])^2 + 2 a (F1[R] + F2[R]) (Derivative[1][F1][R] + 2 Derivative[1][F2][R])*)
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