Symbolic differentiation is a classic problem that illustrates the basic steps of problem reduction search.
Primitive subproblems:
d/dx(x) = 1 d/dx(c) = 0
Subproblem reductions:
d/dx(u + v) = d/dx(u) + d/dx(v) d/dx(u * v) = u * d/dx(v) + v * d/dx(u)In the case of d/dx(u + v), the simpler subproblems are the two smaller derivatives d/dx(u) and d/dx(v). The method of combining subproblem solutions is to add these two results ( algebraically, not numerically).