# 偏微分備忘
偏微分備忘
偏微分とは
ある1つの変数に着目することで、他の変数を固定したときのある変数の変化を導き出すこと
練習
No.1
$$\begin{aligned}
f(x,y) &= x^2 - 3y^3
\\
\dfrac{\partial f}{\partial x} &= 2x
\end{aligned}$$
No.2
$$\begin{aligned}
f(x,y) &= x^2 - 3y^3
\\
\dfrac{\partial f}{\partial y} &= -9y^2
\end{aligned}$$
No.3
$$\begin{aligned}
f(x,y) &= 2x^3 - y + 4xy
\\
\dfrac{\partial f}{\partial x} &= 6x^2 + 4y
\\
\dfrac{\partial f}{\partial y} &= -1 + 4x
\end{aligned}$$
No.4
$$\begin{aligned}
f(x,y,z) &= \sin(x+2y+z^2)
\\
u &= x + 2y + z^2とおくと
\\
f(x,y,z) &= \sin u
\end{aligned}$$
合成関数の微分より$\dfrac{dy}{dx} = \dfrac{dy}{du}\cdot\dfrac{du}{dx}$なので
$$\begin{aligned}
\dfrac{\partial f}
{\partial z} &= \cos u \cdot 2z
\\
&= 2z \cos(x + 2y + z^2)
\end{aligned}$$
No.5
$$\begin{aligned}
\epsilon(a,b)
&:= \sum_{i=1}^{N}(y_i - ax_i - b)^2
\\
\dfrac{\partial \epsilon(a,b)}{\partial a}
&= \sum_{i=1}^{N}2(y_i - ax_i - b)(- x_i)
\\
&= -2 \sum_{i=1}^{N}x_i(y_i - ax_i - b)
\\
\dfrac{\partial \epsilon(a,b)}{\partial b} &= \sum_{i=1}^{N}2(y_i - ax_i - b)(- 1)
\\
&= -2 \sum_{i=1}^{N}(y_i - ax_i - b)
\end{aligned}$$
参考文献
ヨビノリ様動画『【大学数学】偏微分とは何か【解析学】』 https://www.youtube.com/watch?v=UWFTIEIruyc
ヨビノリ様動画『【大学数学】最小二乗法(回帰分析)【確率統計】』 https://www.youtube.com/watch?v=Zz1sgYxrA-k
