Does a functional have to be fed with the whole graph of a function?

Question

Definition of functional goes like a map from vector space to field. If I consider vector space to be function space with single-variable-functions and field to be numbers then is it possible that derivative of a function evaluated at 0 (say) is a functional? This is because the input to a functional is a function, it is the whole graph of the function that should be fed to it( or am I incorrect?) and output of this functional will be a number but for derivative, in this case, we just need the neighbourhood of the function about zero. Is the derivative a functional? Or the definition of funcational correct?