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Take an arbitrary function [tex]f(x)[/tex]. [tex]f(x)dx[/tex] can be thought of an infinitesimal area of a certain form (I emphasise this because I use it later in the argument) determined by the form of the function [tex]f(x)[/tex]. Lets denote its integral by [tex]Y[/tex].

[tex]\int{ f(x)} dx = Y[/tex]

Now, I argue that an infinitesimal of the whole formed by summing up infinitesimals of a certain form must be an infinitesimal of the same form. ie,

[tex] d\int{ f(x)} dx = f(x)dx[/tex]

Then the fundamental theorem of calculus follows.

[tex] f(x) dx = dY [/tex]

[tex] f(x) = \frac{dY}{dx} [/tex]

[tex] f(x) = \frac{d}{dx}\int{ f(x)} dx[/tex]

If this argument is valid. Can it be made rigorous?