I have read a few sources regarding the chain rule, and a pervasive explanation that most of the sources share is this, which is way to sort of make sense of it: "Regard du/dx as the rate of change of u with respect to x, dy/du as the rate of change of y with respect to u, and dy/dx as the rate of change of y with respect to x. If u changes twice as fast as x and y changes three times as fast as u, then it seems reasonable that y changes six times as fast as x, and so we expect that dy/dx = dy/du * du/dx." I don't understand why it is a simple operation of multiplication to find how fast y changes compared to x. Maybe I am missing something. I'd like to mention, though, that I do understand the chain rule; but when I read this description of it, I just don't seem to understand.