method of least square proof

Question

$$Q(\beta_0, \beta_1) = \sum_{i=1}^{n} (Y_i - \beta_0 - \beta_1X_i)^2$$

I'm trying to understand the proof in my notes and it does this

$$\frac{dQ(\beta_0, \beta_1)}{d\beta_0} = 2 \sum_{i=1}^{n} (Y_i - \beta_0 - \beta_1X_i)$$

Shouldn't it be

$$\frac{dQ(\beta_0, \beta_1)}{d\beta_0} = (2)(-1) \sum_{i=1}^{n} (Y_i - \beta_0 - \beta_1X_i)$$

since $(Y_i - \beta_0 - \beta_1 X_i) = (0 - 1 + 0) = (-1)$

Answer 1

I'd agree:$$\frac{\partial}{\partial \beta_0}\sum_i(Y_i-\beta_0-\beta_1X_i)^2=\sum_i\frac{\partial}{\partial \beta_0}(\beta_0+\beta_1X_i-Y_i)^2\\=2\sum_i(\beta_0+\beta_1X_i-Y_i)=-2\sum_i(Y_i-\beta_0-\beta_1X_i).$$My guess is it's a misprint resulting from the fact that we'll set the derivative to $0$, so for any $k\ne0$ $$\frac{\partial}{\partial \beta_0}\sum_i(Y_i-\beta_0-\beta_1X_i)^2=0\iff k\sum_i(Y_i-\beta_0-\beta_1X_i)=0.$$