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I'm implementing (for learning purposes) a logistic regression model.

I've followed this guide.

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 $1$?

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