I'm wondering how can I minimally prove that the dual variable of a linear program (LP) is the derivative of its optimal value w.r.t. to the RHS constants $b$.
$$ Maximize \space c^Tx $$ $$ s.t. \space Ax ≤ b, x ≥ 0; $$
My understanding of the dual variables is from duality of LP from wikipedia, and rather limited to the natural language interpretation therein.
I saw that it is mentioned here, that
The interpretation of the dual variables as derivatives of the optimal value of the objective function with respect to the elements of the right-hand-side is well known in mathematical programming. This result can be extended to ...
How can this be proved in an elementary way (with a minimum number of citation of other necessary theorems)?
I understand the form the dual problem and the formal definition of dual variables. But I have trouble envisioning the derivative of optimal objective value, since it is not a closed form linear function (but rather a linear function subject to a set of linear inequalities).
Is it possible to convert the optimal solution to a closed form function of the RHS, and then take the derivative using linear algebra? (I guess not).