$$
x_i^2
$$\[ x_i^2 \]
$$
\log_2 x
$$\[ \log_2 x \]
$$
10^{10}
$$\[ 10^{10} \]
$$
\{1+2\}
$$\[ \{1+2\} \]
$$
\frac{1+1}{2}+1
$$\[ \frac{1+1}{2}+1 \]
$$
\sum_1^n
$$\[ \sum_1^n \]
$$
\int_1^n
$$\[ \int_1^n \]
$$
lim_{x\to\infty}
$$\[ lim_{x\to\infty} \]
$$
\begin{matrix}
1 & x & x^2 \\
1 & y & y^2 \\
1 & z & z^2 \\
\end{matrix}
$$\[ \begin{matrix} 1 & x & x^2 \\ 1 & y & y^2 \\ 1 & z & z^2 \\ \end{matrix} \]
$$
h(\theta) = \sum_{j=0}^n\theta_jx_j
$$\[ h(\theta) = \sum_{j=0}^n\theta_jx_j \]
$$
\frac{\partial J(\theta)}{\partial\theta_j} = -\frac{1}{m}\sum_{i=0}^m(y^i-h_\theta(x^i))x_j^i
$$\[ \frac{\partial J(\theta)}{\partial\theta_j} = -\frac{1}{m}\sum_{i=0}^m(y^i-h_\theta(x^i))x_j^i \]
$$
f(n) =
\begin{cases}
n/2, & \text{if $n$ is even} \\
3n+1, & \text{if $n$ is odd}
\end{cases}
$$\[ f(n) = \begin{cases} n/2, & \text{if $n$ is even} \\ 3n+1, & \text{if $n$ is odd} \end{cases} \]
$$
\left\{
\begin{array}{}
a_1x+b_1y+c_1z = d_1\\
a_2x+b_2y+c_2z = d_2\\
a_3x+b_3y+c_3z = d_3
\end{array}
\right.
$$\[ \left\{ \begin{array}{} a_1x+b_1y+c_1z = d_1\\ a_2x+b_2y+c_2z = d_2\\ a_3x+b_3y+c_3z = d_3 \end{array} \right. \]
$$
X = \left(
\begin{matrix}
x_{11} &x_{12}&\cdots&x_{1d}\\
x_{21} &x_{22}&\cdots&x_{2d}\\
\vdots&\vdots&\ddots&\vdots\\
x_{m1}&x_{m2}&\cdots&x_{md}
\end{matrix}
\right)
= \left(
\begin{matrix}
x_1^T\\
x_2^T\\
\vdots\\
x_m^T\\
\end{matrix}
\right)
$$\[ X = \left( \begin{matrix} x_{11} &x_{12}&\cdots&x_{1d}\\ x_{21} &x_{22}&\cdots&x_{2d}\\ \vdots&\vdots&\ddots&\vdots\\ x_{m1}&x_{m2}&\cdots&x_{md} \end{matrix} \right) = \left( \begin{matrix} x_1^T\\ x_2^T\\ \vdots\\ x_m^T\\ \end{matrix} \right) \]
$$
\begin{align}
\frac{\partial J(\theta)}{\partial \theta_j}
& = -\frac{1}{m}\sum_{i=0}^m(y^i-h_\theta(x^i))\frac{\partial}{\partial\theta_j}(y^i-h_\theta(x^i)) \\
& = -\frac{1}{m}\sum_{i=0}^m(y^i-h_\theta(x^i))\frac{\partial}{\partial\theta_j}(\sum_{j=0}^n\theta_jx_j^i-y^i) \\
& = -\frac1m\sum_{i=0}^m(y^i-h_\theta(x^i))x_i^j
\end{align}
$$\[ \begin{align} \frac{\partial J(\theta)}{\partial \theta_j} & = -\frac{1}{m}\sum_{i=0}^m(y^i-h_\theta(x^i))\frac{\partial}{\partial\theta_j}(y^i-h_\theta(x^i)) \\ & = -\frac{1}{m}\sum_{i=0}^m(y^i-h_\theta(x^i))\frac{\partial}{\partial\theta_j}(\sum_{j=0}^n\theta_jx_j^i-y^i) \\ & = -\frac1m\sum_{i=0}^m(y^i-h_\theta(x^i))x_i^j \end{align} \]