Whoa! Think of what you’ve accomplished here: You learned enough calculus to find a minima for an equation with two variables, then applied that to the equation for the ∑ϵ2∑ϵ2. You derived the error minimizing values for ^αα^ and ^ββ^, then used those formulae in
R to calculate by hand" the OLS regression for a small dataset.
Congratulate yourself – you deserve it!