Derivative of an Euclidean norm of vectors

Question

In his Convex Optimization book, Boyd has a proof for the Separating hyperplane theorem. In page 48 in particular there is this part that I am not sure how it is done,

How did we end up from the derivative to the right part of the equation?

Answer 1

Note that $||x||^2 = \langle x,x \rangle = x^T x$ and that the scalar product is bilinear. So, by the product rule,

$$\frac{d}{dt}\langle v+tw ,v\rangle|_{t=0} = \langle w,v \rangle + \langle v,w \rangle = 2 \langle w,v \rangle$$

(if you have problems with taking the derivative of a sum of vectors write it down in coordinates).

Now apply this to your special case.