Most people prefer to prove it from the chain rule alone, which doesn't depend on the fractional notation.
Just in case a picture helps...
... is the chain rule. Straight continuous lines differentiate downwards (integrate up) with respect to the main variable (in this case x), and the straight dashed line similarly but with respect to the dashed balloon expression (the inner function of the composite which is subject to the chain rule).
From the bottom row, dx/du = one over du/dx.
See here for an interesting historical perspective on (and reaction against) nervousness about taking the fractional notation literally...
Non-standard analysis - Wikipedia, the free encyclopedia
Don't integrate - balloontegrate!
Balloon Calculus; standard integrals, derivatives and methods
Balloon Calculus Drawing with LaTeX and Asymptote!