  We give an elementary approach utilizing only the divided difference
formalism for obtaining expansions of Schubert polynomials that are manifestly
nonnegative, by studying solutions to the equation $\sum
Y_i\partial_i=\mathrm{id}$ on polynomials with no constant term. This in
particular recovers the pipe dream and slide polynomial expansions. We also
show that slide polynomials satisfy an analogue of the divided difference
formalisms for Schubert polynomials and forest polynomials, which gives a
simple method for extracting the coefficients of slide polynomials in the slide
polynomial decomposition of an arbitrary polynomial.
