Hi I know that the common explanation of derivatives is drawing a secant line through a graph and move one point closer to the other where the space between them is infinitesimal. Similarly, area under a graph can be found by finding the areas of individual rectangles with infinitesimally small width and adding the rectangles' areas together. But I've been reading some material that was assigned by my professor that explains the paradoxes in these common explanations. Ex. as we move two points closer and closer together eventually we would get 0/0 as the slope. Can anyone come up with an explanation that avoids such paradoxes?