Let suppose we have the following vector
$\ \mathbf{b}= [b_1, b_2, b_3]$
$\:R_j\left(b_j\right)=\dfrac{b_j Q^2}{\sum _{i=1}^3\left(b_i\right)\:}-\dfrac{b_j Q}{\sum_{i=1}^3\left(b_i\right)\:} $
$ \dfrac{\partial R_j\left(\mathbf{b}\right)}{\partial b_j}\:=?\: $
How to differentiate?
There is no problem at all; you just have to set up things correctly. You have the three functions $$R_j({\bf b}):={b_j\over\sum_i b_i}\>(Q^2-Q)\qquad(1\leq j\leq3)\ ,$$ whereby $Q$ is assumed constant. Then $${\partial R_j\over\partial b_j}={1\cdot\sum_i b_i-b_j\cdot 1\over\left(\sum_i b_i\right)^2}\>(Q^2-Q)={\sum_{i\ne j} b_i\over\left(\sum_i b_i\right)^2}\>(Q^2-Q)\ .$$ When $k\ne j$ we similarly have $${\partial R_j\over\partial b_k}={0\cdot\sum_i b_i-b_j\cdot 1\over\left(\sum_i b_i\right)^2}\>(Q^2-Q)={-b_j\over\left(\sum_i b_i\right)^2}\>(Q^2-Q)\qquad(k\ne j)\ .$$