# Proof in regression model

Im studying for an exam, i don't have the solution, so I hope some of you guys can help me. I have tried a lot but i can't do this proof.

Suoppose we have the linear regression model $$U_i=\beta_kX_{k,i}+u_i,$$ $$i=1,2 .... , n,$$
where $$k$$ is an index which we will use later ( it will be the k´th covariate in a more general linear regression model). The least squares estimator of $$\beta_k$$, which we call for $$\hat{b}_k$$ is the minimizer of
$$L_k(b_k)= \sum_{i=1}^{n}(Y_i-b_kX_{k,i})^2$$.
Show that the first order equation $$\frac{d}{db_k}L_k(b_k)=0$$ has the single solution $$\hat{b}=\frac{\sum ^n_{i=1}X_{k,i}Y_k }{\sum ^n_{i=1}X^{2}_{k,i}}$$
You may without justification conclude that this is the minimizer of $$L_k(b_k)$$
• @OP: Did you try taking the derivative: $\frac{\mathrm{d}}{\mathrm{d}b_k}\sum_{i=1}^{n}(Y_i-b_kX_{k,i})^2$? – Winter Soldier May 9 at 19:43