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I'm kind of confused about the rules for partially differentiating summations...what would the partial derivative of this be wrt to beta? Thank you!

$\frac{\partial}{\partial \beta} \sum_{i=1}^n -2x_iy_i+2\beta x_i^2$

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The rules for partial differentiation are the same as regular differentiation, just treat all other independent variables as constants, so the result will be $$\sum_{i=1}^n2 x_i^2$$As long as $x_i,y_i$ are independent variables or constants.

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