Re-express A2, B2, and C2 as, respectively,

where simply denote the change in each original quantity.

Your "post-change" expression can then be rendered as

Multiplying out the LHS gives you

Finally deduct your first equation from this last one, leaving

In other words, theinchangeCis the sum of three influences: The change inoperating on the 'original'A; the change inBoperating on the 'original'B; and the "cross product" ofAA's andB's change.

There's a geometric interpretation of your question...

...think of youroriginalquantity C1 as the green shaded area, it having an area equal to the product of A1 and B1 (with A1 and B1 playing the roles of length and width). Then length and width change by some quantities, resulting in the larger rectangle. Your question amounts to asking what gives rise to thechangein the area of the rectangle, and thisarea-changeis the yellow-shaded area. You can see that the yellow area is the sum of three components, each one corresponding to those three 'influences' in my final equation.