Multiplicative derivative (${\partial }_{h}f(x)=f(x+h)\overline{f(x)}$)

A product that cancels constant phases and lowers the degree of a polynomial phase. For $f(x)=e(\alpha x^2)$, it is $e(2\alpha hx+\alpha h^2)$. Some conventions conjugate the entire expression and reverse the frequency signs. The Gowers uniformity norm derivative identity relates a higher norm to the average of lower norms of these derivatives.