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real function argument

I have a problrm with defining real argument

In:   f[x_]:=Re[Cos[x]+I Sin[x]]

Out:  -Cos[x] Im'[x]-Sin[x]Re'[x]

In: FullSimplify[f'[x],Element[x,Reals]] 
gives the same out

Any suggestions ?

Regards, Olaf

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