I get the point of the author about differentiation and integration. But, I did not follow completely how he connected it to analysis and synthesis. I mean, there isn't necessarily a one-to-one mapping between analysis and differentiation and synthesis and integration, right?
Maybe something like: When you do derivation you analyze how the function behaves at a given point. Whenever you analyze something you analyze a given thing, not all things. When you do integration you look at a function and then produce something that describes the function on every point. You come up with a closed integral-formula for the shole function. You need to synthesize something that describes the function everywhere. But when you derivate the dericate, you also produce a closed function that describes the function everywhere. Oops, I think I got lost here too.