I derive the least squares estimators of the slope and intercept in simple linear regression (Using summation notation, and no matrices.) I assume that the viewer has already been introduced to the linear regression model, but I do provide a brief review in the first few minutes. I assume that you have a basic knowledge of differential calculus, including the power rule and the chain rule.
If you are already familiar with the problem, and you are just looking for help with the mathematics of the derivation, the derivation starts at 3:26.
At the end of the video, I illustrate that sum(X_i-X bar)(Y_i - Y bar) = sum X_i(Y_i - Y bar) =sum Y_i(X_i - X bar) , and that sum(X_i-X bar)^2 = sum X_i(X_i - X bar).
There are, of course, a number of ways of expressing the formula for the slope estimator, and I make no attempt to list them all in this video.