$$
x_i^2
$$\ [X_I ^ 2 \]
$$
\log_2 x
$$\ [\ log_2 X \]
$$
10^{10}
$$\ [10 ^ {10} \]
$$
\{1+2\}
$$\ [\ {1 + 2 \} \]
$$
\frac{1+1}{2}+1
$$\ [\ FRAC {1 + 1} {2} +1 \]
$$
\sum_1^n
$$\ [\ sum_1 ^ N \]
$$
\int_1^n
$$\ [\ int_1 ^ N \]
$$
lim_{x\to\infty}
$$\ [lim_ {X \ inftyのに\} \]
$$
\begin{matrix}
1 & x & x^2 \\
1 & y & y^2 \\
1 & z & z^2 \\
\end{matrix}
$$\ [\ \端{行列} \ {行列} ^ 2 \\ 1・X・X ^ 2 \\ 1＆Y＆Y ^ 2 \\ 1＆Z＆Zを開始します]
$$
h(\theta) = \sum_{j=0}^n\theta_jx_j
$$\ [H（\シータ）= \ sum_ {J = 0} ^ N \ theta_jx_j \]
$$
\frac{\partial J(\theta)}{\partial\theta_j} = -\frac{1}{m}\sum_{i=0}^m(y^i-h_\theta(x^i))x_j^i
$$\ [\ FRAC {\部分J（\シータ）} {\部分の\ theta_j} = - \ FRAC {1} {M} \ sum_ {i = 0} ^ M（Y ^ I-H_ \シータ（X ^ I ））X - jが^私は\]
$$
f(n) =
\begin{cases}
n/2, & \text{if $n$ is even} \\
3n+1, & \text{if $n$ is odd}
\end{cases}
$$\ [F（N）= \ {$ N $が偶数の場合}、{$ N $が奇数の場合} 3N + 1、＆\テキスト\\ \テキスト＆2 / N {ケースを}開始\端{ケース} \ ]
$$
\left\{
\begin{array}{}
a_1x+b_1y+c_1z = d_1\\
a_2x+b_2y+c_2z = d_2\\
a_3x+b_3y+c_3z = d_3
\end{array}
\right.
$$\ [\左\ {\端{アレイ} \右\ {アレイ} {} a_1x + b_1y + c_1z = D_1 \\ a_2x + b_2y + c_2z = D_2 \\ a_3x + b_3y + c_3z = D_3を始めます。\]
$$
X = \left(
\begin{matrix}
x_{11} &x_{12}&\cdots&x_{1d}\\
x_{21} &x_{22}&\cdots&x_{2d}\\
\vdots&\vdots&\ddots&\vdots\\
x_{m1}&x_{m2}&\cdots&x_{md}
\end{matrix}
\right)
= \left(
\begin{matrix}
x_1^T\\
x_2^T\\
\vdots\\
x_m^T\\
\end{matrix}
\right)
$$\ [X = \（左\開始{行列} X_ {11}＆X_ {12} - \ cdots＆X_ {1D} \\ X_ {21}＆X_ {22} - \ cdots＆X_ {2D} \\ \ vdots＆\ vdots＆\ ddots＆ \ vdots \\ X_ {M1}＆X_ {M2}＆\ cdots＆X_ {MD} \端{行列} \右）= \左（\ {行列} X_1を開始^ T \\ X_2 ^ T \\ \ vdots \\ x_m ^ T \\ \端{行列} \右）\]
$$
\begin{align}
\frac{\partial J(\theta)}{\partial \theta_j}
& = -\frac{1}{m}\sum_{i=0}^m(y^i-h_\theta(x^i))\frac{\partial}{\partial\theta_j}(y^i-h_\theta(x^i)) \\
& = -\frac{1}{m}\sum_{i=0}^m(y^i-h_\theta(x^i))\frac{\partial}{\partial\theta_j}(\sum_{j=0}^n\theta_jx_j^i-y^i) \\
& = -\frac1m\sum_{i=0}^m(y^i-h_\theta(x^i))x_i^j
\end{align}
$$\ FRAC {1} {M} \ sum_ {i = 0} ^ M（Y ^ I-H_ - \ [\ \ {整列} FRAC {\部分J（\シータ）} {\部分の\ theta_j}＆=を開始\シータ（X ^ i））を\ FRAC {\部分} {\部分の\ theta_j}（y ^ I-H_ \シータ（X ^ i））を\\＆= - \ FRAC {1} {M} \ sum_ { I = 0} ^ M（Y ^ I-H_ \シータ（X ^ i））を\ FRAC {\部分} {\部分の\ theta_j}（\ sum_ {J = 0} ^ N \ theta_jx_j ^ IY ^ I）\ \＆= - \ frac1m \ sum_ {i = 0} ^ M（Y ^ I-H_ \シータ（X ^ i））をX_I ^ j個の\端{整列} \]