What's your favourite example of where it doesn't work? Physics is full of quasi-infinistesimal quantities and I always like good counter examples (ideally without invoking something like the blamange function or similar....)
Where this doesn't work is if you have more than 1 dimension. Then you need to deal with the added complexity of integrating modular forms and the fact that in 2D you don't have df = (df/dx) dx but df = (df/dx) dx + (df/dy) dx. The chain rule also changes into a matrix product, rather than a simple dz/dx = dy/dx dz/dy.