When we know the optimal solution of a function, we need to solve for the derivative of the value function of this function with respect to a certain parameter in the function, then we can use the envelope theorem.
1. Envelope theorem under unconstrained conditions
The result of a function taking a partial derivative of a parameter (a parameter that is not an independent variable) at its extreme point is equivalent to the result of the function taking a partial derivative of its parameter and then bringing it into the extreme point.
2. Envelope theorem under constrained conditions
For the restrictions with constraint g(x,a), we can get the following:
