[學習] Implicit function theorem

還有一個強大的定理 Implicit function theorem
基本上就是想知道 dy/dx 但是所有的關係都是透過另一個函數連在一起的
所以有人就很聰明的發展這個定理
就算兩個變數是沒有直接相關連的
我們也可以很快知道彼此對應的影響
推導的內容也只是用到簡單的chain rule而已
但是卻真的滿好用得
記錄一下

http://en.wikipedia.org/wiki/Implicit_function_theorem

[學習] Envelope theorem

記錄一下寫作業的時候用到的兩個定理：
Envelope theorem:
當面對 min f(x,r)
想知道單一變數偏微的效果時可以使用(以外為引用沒有限制的情況下)

Consider an arbitrary maximization (or minimization) problem where the objective function f(x,r) depends on some parameters r:

The function f *(r) is the problem's optimal-value function — it gives the maximized (or minimized) value of the objective function f(x,r)as a function of its parameters r.

Let x*(r) be the (arg max) value of x, expressed in terms of the parameters, that solves the optimisation problem, so that f*(r)=f(x*(r),r). The envelope theorem tells us how f*(r) changes as a parameter changes, namely:

That is, the derivative of f*(r) with respect to r_i is given by the partial derivative of f(x,r) with respect to r_i, holding x fixed, and then evaluating at the optimal choice x=x*(r).

http://en.wikipedia.org/wiki/Envelope_theorem
應該是要求 f is twice differentiable continuous function.