I am running into a real world problem. But I think this is more like a math problem. So here it is. Suppose I have $$A = B + C.$$
The $A, B, C$ in this period are called $A_{1}, B_{1}, C_{1}$. Similarly, $A, B, C$ in previous period are given by $A_{0}, B_{0}, C_{0}$. The changes in $A$ compared to the previous period is give by $$\Delta A = \frac{A_{1}}{A_{0}}-1.$$
The changes in $B$ and $C$ are defined in a similar way. I want to decompose the changes in $A$ in to changes in $B$ and $C$. Obviously, I cannot do the following $$\Delta A = \Delta B + \Delta C$$ since mathematically it would be wrong.
I wonder if there is a smarter way to decompose $\Delta A$ into changes in $B$ and changes in $C$, which makes sense regarding the restriction $A = B + C$ and is also mathematically correct. Maybe there's not such a way. But just curious.
Using your notation for $\Delta A$ (which most people would call $\frac{\delta A}{A}$), the formula you are looking for is
$$ \Delta A = \frac{1}{A_0}\left( B_0 \Delta B + C_0 \Delta C \right) $$