The pure algebraic proof is more complex and requires consideration of the system equation:
where  is the stoichiometry matrix and  the rate vector. In this derivation, we assume there are no conserved moieties in the network, but this doesn't invalidate the proof. Using the chain rule and differentiating with respect to  yields, after rearrangement:
The inverted term is the unscaled control coefficient so that after scaling, it is possible to write:
To derive the flux response coefficient theorem, we must use the additional equation: