Let $E$ be an elliptic curve defined over a field $k$ given by Weierstrass equation
$$y^2+a_1xy+a_3y=x^3+a_2x^2+a_4x+a_6\ .$$
Every standard book of elliptic curve includes the formula for discriminant $\Delta$ as $$ \begin{aligned} \Delta &=-b_2^2b_8 - 8 b_4^3 -27 b_6 ^2 + 9 b_2 b_4 b_6\ , \text{ where}\\[3mm] b_2 &= a_1^2 + 4 a_2\\ b_4 &= 2a_4 + a_1 a_3\\ b_6 &= a_3^2 + 4 a_6 \\ b_8 &= a_1^2 a_6 + 4 a_2 a_6 - a_1 a_3 a_4 + a_2 a_3^2 - a_4^2. \end{aligned} $$
But I would like to find the derivation of these quantities which I could not find in any of these books. It will be very helpful if I can get the derivation or any hints or any reference which includes the derivation.
Thanks in advance for your help.