I have 3 sets called A,B and C. Each set is a union of 2 non intersecting subsets, i.e. :

$\displaystyle A = \{A1 \cup A2\}, A1 \cap A2 = \emptyset$

$\displaystyle B = \{B1 \cup B2\}, B1 \cap B2 = \emptyset$

$\displaystyle C = \{C1 \cup C2\}, C1 \cap C2 = \emptyset$

I'm interested in knowing what happens to the quantity

|A| / (|A| + |B| + |C|)

when I change :

|A2| / (|A2| + |B2| + |C2|)

One way might be to take the derivative of the first quantity with respect to the second quantity, although it doesn't seem to be an easy computation ? Is there another way ?