Fit $y=\beta_0+\beta_1 x$: 寫成 matrix form 比較好計算:
$$ \left[ \begin{array}{cc} n & \sum x_i \\ \sum x_i & \sum x_i^2 \end{array} \right] \left[ \begin{array}{c} \beta_0 \\ \beta_1 \end{array} \right] = \left[ \begin{array}{c} \sum y_i \\ \sum x_i y_i \end{array} \right] $$
Total derivative 和 partial derivative 關係式 for $f=f(x,y)$:
$$ df=\frac{\partial f}{\partial x}dx + \frac{\partial f}{\partial y}dy $$
由此可推導出 chain rule
讓人意外的是這個式子 (triple product rule):
$$ \frac{\partial x}{\partial y}\frac{\partial y}{\partial z}\frac{\partial z}{\partial x}={\color{orange}{-1}} $$
這是建立在如果有這樣的關係: $f(x,y,z)=0$.
參考 wiki: https://zh.wikipedia.org/wiki/三乘积法则
注釋說的是要注意的條件: “每一個變量可視作另外兩個變量的函數”
https://en.wikipedia.org/wiki/Implicit_function#In_differential_equations