Given $v = v_1 + v_2 \in \mathbb{R}^n$, we know that:
$$ \frac{\partial}{\partial v} \big(v^\top v\big) = \frac{\partial}{\partial v} \big(v_1^\top v_1 + v_2^\top v_2 + 2v_1^\top v_2\big)= 2v$$
so then, really, what are: $$ \frac{\partial}{\partial v} \big(v_1^\top v_1\big), \ \ \frac{\partial}{\partial v} \big(v_2^\top v_2\big) \ \ \text{ and } \ \ \frac{\partial}{\partial v} \big(2v_1^\top v_2\big) = ? $$
I'm sure they can't be zero. It should be very simple, but I can't seem to grasp what's going on once I look at each of these terms individually and knowing that $v = v_1+v_2$. Thank you in advance.
They're related by $v_1=v-v_2$. Thus if $v_2$ is independent of $v$, then $v_1$ becomes a function of $v$ and vice versa to keep the derivative as it is.