I did not fully understand the intention, it looks like you are trying to assign a function to an operator. If you use the name df the following works:

df[x_] := Sin[x]
(* save *)
Export["/tmp/test.wdx", df[x]]
(* clear *)
Clear@df
(* load *)
df[x_] := Evaluate@Import["/tmp/test.wdx"]

This works for me:

ClearAttributes[Derivative, {ReadProtected}]
Derivative[1][f][x_] := Sin[x]
Save["file.m", {f, Derivative}]

Edit

Included the line that clears the ReadProtected attribute, in case it's necessary. It seems like this might depend on version or something, because for me Derivative never has that attribute in a new kernel.