Can differentiation be thought of as a function?$ $

Solution 1:

Yes, differentiation can be thought of as a function from the set of differentiable functions to the set of functions which are derivatives of a differentiable function. (which is, as Dr. MV points out in a comment, not quite the set of integrable functions).

Such things, which map functions to functions, are typically called operators, but this is just a convention, you can think of them as functions just fine.

Solution 2:

Differentiation of a function is a linear operator, a function on a set of functions.

Differentiation at a point is a linear functional, a function which maps elements of a vector space to it's underlying field, in this case functions to real numbers.

Solution 3:

Look up differential forms. I think you'll find that what you're describing closely resembles them. In fact, any course on differential topology or differential geometry should present this perspective.