Are you sure you got the point, here? 'Flipping' is involved, maybe, but in order to cancel the three-over-two fraction that comes as the derivative of 'three over two u', which is the inner function of a chain rule process.
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