Were the properties of them discovered together or was it a separate discovery?

Were the properties of them discovered together or was it a separate discovery?

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https://www.math.ucla.edu/~tao/preprints/forms.pdf might be of interest.

I'm sure that Terry Tao exposition is very good, but I think there's a simpler answer.

Imagine a simple function: f(x) = 2. The area under that graph from 0 to x is a rectangle, of course. The question to think about is: how quickly does the area grow. Well, if the base gets one unit wider, the area of the rectangle increases by 2. If the base gets 1/2 of a unit wider, the area of the rectangle increases by 1. If the base increases by a tiny amount dx, the area increases by 2 * dx. That is, dA = 2dx, or dA/dx = 2. I.e. the derivative of the area (with respect to the base x) is 2, exactly the value of the function that we started with.

This is true in general - if we've found the area under the graph from 0 to x and we want to widen the area just a bit more, we can increase the width by a tiny amount dx. How much area will this add? Well, it's basically like tacking on a tiny little rectangle of width dx and height equal to the height of the function at that point. So the change in area is f(x) * dx. Or, rewriting that, dA = f(x) * dx, so dA/dx = f(x). The derivative of the area is f(x). That's why the derivative of the integral is the original function.

It depends on what you mean specifically by integration and antidifferentiation, i.e., the exact definitions you're working with.

Without any other context, I would just say that they are synonyms, which is why they're the same.

They're just inverse processes, like adding vs subtracting. Differentiation finds the rate of change of a function. Integration find the set of functions with a given rate of change.

Edit: I don't know the details of their discovery but I assume the concept of inverses was fairly obvious to Newton and Leibniz.

OP is asking about integration and antidifferentiation.

I thought that's what I explained? Both are the inverse of differentiation, hence they are the same?

Sorry, I don't see it. You wrote "They are inverse processes", but they aren't. Differentiation and integration are, so that's what it looks like you are referring to.

Integration find the set of functions with a given rate of change.

yes, OP is asking why this is true. your comment doesn't answer the question.

## Youre_Government

What you're asking about is literally called the Fundamental Theorem of Calculus. It's not trivial, and I doubt it was all realized in one fell swoop.