I am a little confused over part 1 of the fundamental theorem of calculus. Part 2 makes perfect sense to me. I guess my confusion is if we have an integral g(x) defined from [a, b], and we are looking at point x, how do we know that g'(x) = f(x)? It makes sense in the idea that they are inverses. I understand that much. One "undoes" the other. I guess my confusion spawns from this idea that only the upper limit being a variable affects anything. It seems like the lower limit should be considered. Do we ignore it because it's a constant, so whenever we take the derivative it will just be 0? I don't really know where my confusion is. I think it's because the idea kind of seems so obvious that I'm over thinking it. That, and we covered Part 2 first, which seemed to kind of make Part 1 redundant since in Part 2 we were doing applications where if we had(adsbygoogle = window.adsbygoogle || []).push({});

G(x) = ∫ G'(x) dx, which seems to flow naturally that since an integral is an antiderivative, the inner function must be a derivative of the integral.

Also, in my book, they talk about using the chain rule if your upper limit is something like x^{2}. Is this because you have f(g(x)), where g(x) = x^{2}so when you do part 1 of the fundamental theorem you have f'(g(x))g'(x)? The book doesn't explain this part. It just says to apply the chain rule and gives an anwser, but it's a tad confusing since we've never applied the chain rule to an integral before this.

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# Fundamental Theorem of Calculus: Part One

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