Vector Algebra + Partial Derivative

[nosit]

Hi!

I am not sure if this forum is the appropriate place to post this question, but here it goes...

This is about OLS (Ordinary Least Squares Estimator), where we choose a slope in such a way that I minimize the squared residuals.

In order to minimize this function, the professor puts a derivative with respect to Beta (the slope) equals to zero. So far so good.

In the step I have put in yellow, I see he applies the Chain Rule, so 2 is behind the Summation symbol due to power rule and then he derived the inner function.

The problem is that I don't understand why the result is Xi and not Xi' (which means X transposed). Does anyone know this?

 

[Dr.Peterson]

It may help if you explain a little more of the context. What does [MATH]x'_i[/MATH] mean? (You call it a transpose; does that mean that [MATH]x_i[/MATH] is a vector?) Also, how does [MATH]\beta[/MATH] in line 1 become [MATH]\hat{\beta}[/MATH] in line 2?

 

[HallsofIvy]

If you are differentiating z= f(y+ x^2) with respect to x, y independent of x, then, since the derivative of x^2 is 2x, z'= 2xf'(y+ x^2).