Suppose we have 3 vectors satisfying this relation (only 2 are independent we take it as $p_1,p_3$.

$vec{p_1}+vec{p_2}+vec{p_3}=0$

We want to derive

$frac{partial vec{p_2}}{partial |p_1|}$

where $|vec{p_1}|equiv sqrt{vec{p_1}^2}$

What is the value of

$frac{partial vec{p_2}}{partial |p_1|}$ in terms of $vec{p_3},vec{p_1}$?