Gradient of squared dot product

Question

Let $x,y \in \mathbb{R}^d$. What is

$$\nabla_x\langle x,y\rangle^2$$

I know that we have

$$ \nabla_x\langle x,x\rangle^2 = \|x\|^2 = 2x $$ but how this changes for the case $\langle x,y\rangle^2$?

Answer 1

We have $$\begin{align}d_p\langle x,y\rangle^2&=2\langle x,y\rangle\langle p,y\rangle\\ &=\langle2\langle x,y\rangle y,p\rangle, \end{align} $$ hence the gradient is $2\langle x,y\rangle y.$