Maybe you did entirely get the point - then forgive me this diagram opportunity...
As usual, straight continuous lines differentiate downwards (/integrate up), the straight dashed line similarly but with respect to the dashed balloon expression - so that the triangular network on the right satisfies the chain rule for differentiation.
The 'flipping' would be, perhaps, the multiplying of the one-over-two fraction by the reciprocal of three-over-two.
Don't integrate - balloontegrate!
Balloon Calculus Forum