I'm reading V.I. Arnold's Huygens and Barrow, Newton and Hooke and he gives this diagram as part of his discussion of the derivation of the product rule, and I can't see what's wrong with it, except that it seems to give a different version of the product rule than the generally accepted one.
Since $dx$ and $dy$ are infinitesimal, their product $dxdy$ is a second order infinitesimal and so can be ignored.
Of course that's not a rigorous argument - Newton didn't need one. It can be made rigorous when preceded by proper definitions.