I am having hard time understanding the concept of double summation derivation. I have this model $${L}=\sum_{i=1}^a\sum_{j=1}^n (y_{ij}-\mu-\tau_i)^2$$ which is the fixed effects model for one way ANOVA. Now, to estimate the parameters $\mu$ and $\tau_i$ I need to used the least squares method and using the partial derivatives I would be able to get them. The book says that $$\frac{\partial{L}}{\partial{\mu}}=-2\sum_{i=1}^a\sum_{j=1}^n (y_{ij}-\hat\mu-\hat\tau_i)=0$$ which I understand. What I don't understand is that the other derivative $$\frac{\partial{L}}{\partial{\tau}}=-2\sum_{j=1}^n(y_{ij}+\hat\mu-\hat\tau_i)=0$$
How does the sign differ between these two derivatives? Also, where did the sigma operator of i go? can you derive the 2nd derivative in simple steps?
Thanks
You missed the fact that what you have to write is the derivative with respect to $\tau_i$ $$\frac{\partial{L}}{\partial{\tau_{\color{red}{i}}}}=-2\sum_{j=1}^n(y_{ij}+\hat\mu-\hat\tau_i)$$ About the sign, don't care since you will set the derivatives equal to $0$.