Now, the author is taking the derivative of $$l$$, the cost function with respect to some $$\beta_j$$:
$$\frac{\partial l}{\partial \beta_j} =\sum_{i = 1}^m y_i x_{ij} - \sum_{i = 1}^m \frac{e^{\beta_0 + \beta \cdot x_i}}{1 + e^{\beta_0 + \beta \cdot x_i}} \cdot x_{ij} \\ = \sum_{i=1}^m x_{ij} (y_i - p(x_i;\beta_0, \beta))$$
my question is regarding $$\beta_0$$. Basically the derivative is different for $$\beta_0$$.
Is that reasonable, for simplicity, to ommit $$\beta_0$$ and instead extending $$x_i$$ to a larger dimension where the first term is a constant of $$1$$?