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This means that f(x), an analytic expression for example, is the image of x under f, and not the function itself, just as y, the output value, is not the function itself. Thus, why do we say "take the derivative of f(x)" when f(x) is not even the function, but rather the image of the variable x under f? In addition, the notation dy/dx gives the feeling that the input to a differentiation operator is not a function at all, but an expression that defines a function, y=f(x). Given that we say "take the derivative of such and such function", shouldn't we always write D(x↦x

^{2})=(x↦2x) rather than D(x

^{2})=2x. Is the latter just a shorthand for the more rigorous former?