In a way, you can think of calculus as using exactly this sort of idea. Say you are interested in determining the area of the region consisting of points (x,y) with $a \le x \le b$ and $0 \le y \le f(x)$ . It turns out the best thing to do is to consider the family of regions $ S(c) $ consisting of points (x,y) with $ a \le x \le c, 0 \le y \le f(x) $, because then you can associate to this family the function A(c) = area of S(c). Then we know A'(c)=f(c), and this is a much easier problem to solve (and then just evaluate at c=b) than, say, trying to calculate the area of a single region directly, say by exhaustion.