I am reading Apostol's Modular Functions and Dirichlet Series in Number Theory. And I am stuck on exercise 8 of Chapter 1. The problem asks to show that $$\wp''\left(\frac{\omega_1}{2}\right)=2(e_1-e_2)(e_1-e_3),$$ where $e_1=\wp\left(\frac{\omega_1}{2}\right)$, $e_2=\wp\left(\frac{\omega_2}{2}\right)$, and $e_3=\wp\left(\frac{\omega_1+\omega_2}{2}\right)$.
I did not know how to approach the problem, so I started by differentiating the differential equation for $\wp$ and get $\wp''(z)=6\wp^2(z)-\frac{1}{2}g_2$. Therefore, $\wp''(e_1)=6e_1-\frac{1}{2}g_2$. Then using the original differential equation, and plugging in $e_1, e_2, e_3$, we have $$g_2e_1+g_3=4e_1^3,$$ $$g_2e_2+g_3=4e_2^3,$$ $$g_2e_3+g_3=4e_3^3.$$ We can solve for $g_2$ and plug it back to get $\wp''(e_1)=6e_1^2-2(e_2^2+e_2e_3+e_3^2)$. I cannot proceed from there.