How important are differentials and linear approximation in the study of calculus? I mean the dy=f'(x)dx stuff. It seems simple but I always thought you couldn't treat the dy/dx as a fraction? And can the integration...formulas be derived without using differentials (think it's the integration by parts one)? I've been taught some methods on how to integrate and I have no idea why I'm doing them, e.g. one where you take the derivative of one of the substitutions and you invert then you multiply it...etc.