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 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 x2. Is this because you have f(g(x)), where g(x) = x2 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.