The parameters for the hypothesis are \(\theta_0,\theta_1,\theta_2,\dotsc,\theta_n\), we can think of those parameters simply as a vector \(\theta \in \mathbb{R}^{n+1} \).
The cost function here is: \(J(\theta_0,\theta_1,\dotsc,\theta_n) = \frac{1}{2m} \sum\limits_{i=1}^m (h_{\theta}(x^{(i)}) - y^{(i)})^2\)
\[\theta_j := \theta_j - \alpha \frac{\partial}{\partial \theta_j} J(\theta_0,\theta_1,\dotsc,\theta_n)\]