Buffer capacity of an aqueous solution

The buffer capacity of an aqueous solution, β_s, consists of the buffer capacities of the components of the solution:

$\beta_s=\beta_w+\beta_a+\beta_b+...\; \; \; \; \; \; \; \; (7)$

where β_w is the buffer capacity of water, β_ais the buffer capacity of a weak acid and β_bis the buffer capacity of a weak base.

To prove that buffer capacities are additive, we substitute eq1 in eq7

$\frac{dc_{b,s}}{dpH}=\frac{dc_{b,w}}{dpH}+\frac{dc_{b,a}}{dpH}+\frac{dc_{b,b}}{dpH}+...$

$\frac{dc_{b,s}}{dpH}=\frac{d\left (c_{b,w}+c_{b,a}+c_{b,b}+... \right )}{dpH}$