Creating a least-squares matrix of partial derivatives

[BifSlamkovich]

In the ordinary least squares procedure I have obtained an expression for the sum of squared residuals, S, and then took the partial derivatives of it wrt β₀ and β₁. Help me to condense it into the matrix, -2X'y + 2X'Xb.

∂S/∂β₀ = -2y₁x₁₁ + 2x₁₁(β₀x₁₁ + β₁x₁₂) + ... + -2y_nx_n1 + 2x_n1(β₀x_n1 + β₁x_n2)

∂S/∂β₁ = -2y₁x₁₂ + 2x₁₂(β₀x₁₁ + β₁x₁₂) + ... + -2y_nx_n2 + 2x_n2(β₀x_n1 + β₁x_n2)

At least, help me to understand why the two partial derivatives equal the matrix above.

 

[HallsofIvy]

Since you haven't told us what "yi" or "xij" are or what they have to do with S, $\beta_0$, or $\beta_1$, it is impossible to answer your question.