- #1
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I have a question concerning how how we define the differentiation and integration operators. Firstly, I know that functions are typically defined as an ordered triple triple ##(X, Y, f)## such that ##f⊆X×Y##, where ##x \in X## and ##f(x) \in Y##. This all seems nice and fine, but we also define the differentiation operator as ##\frac{d}{dx}: f \mapsto f'##. My confusion lies in the fact that ##\frac{d}{dx}## takes expressions such as ##5x^2 + 2 = f(x)## as inputs, not ##f##; e.g., ##\frac{d}{dx} (5x^2 + 1) = 10x## (Obviously, an ordered triple is not the input). The operator seems to take the image of ##x## under ##f##, which is ##f(x)##, not the function ##f## itself, which is the ordered triple. Why, then, do we define the differentiation operator as mapper of functions, when it takes ##f(x)##, an expression, as input, rather than ##f##, a function, as input? This question also applies to the indefinite integral operator, which supposedly maps ##f## to ##F##, where ##F' = f##.