Why does the chain rule work?

One of the best ways to view dy/dx is as a fraction. When we have y=f(g(x)), we need to make a substitution u=g(x) to find dy/dx. This leaves us y=f(u) and u=g(x). Differentiating said terms leaves us with dy/du=f’(u) and du/dx = g’(x).But why does this help us? We’ve just made this more complicated by adding a new variable right? Well, that’s actually not true. If we multiply our two differentiated terms, you should be able to spot that the du terms cancel out (fraction cancellation), and thus we’re left with dy/dx. Then we sub back in our u=g(x). So dy/dx = f’(g(x))g’(x).