TITLE: Can differentiation be thought of as a function?$ $
QUESTION [27 upvotes]: Would it be wrong to think of differentiation as a function whose domain is the set of differentiable functions and co-domain is the set of all functions whose domain and range are some subsets of real numbers?

REPLY [2 votes]: Differentiation can indeed be  thought of as a function, but it was not viewed that way by the creators of the calculus.  The first mathematician to think of differentiation as an operator from functions to functions was Euler's successor Lagrange.