The gradient of the dot product of a point and a vector?

Question

This is a list of vector identities from the first page of the text on Classical Electrodynamics by Jackson. There is not much explanation provided.

https://www.physics.rutgers.edu/~shapiro/504/lects/vecidents.pdf

9. is the usual gradient of a dot product of two vectors identity.

Then: 15. $$ \nabla(\vec A \cdot \vec x) $$

Is $\vec x$ a point? Never encountered this I think? Can someone give me an explanation? I don't even know what's the proper name for this identity.

Thanks

Answer 1

Hint

Let us see the $j^{th}$ component :

$$(\vec{\nabla}(\vec{A}•\vec{x}))_j=$$

$$\frac{\partial \vec{A}•\vec{x}}{\partial x_j}=$$

$$\frac{\partial \vec{A}}{\partial x_j}•\vec{x}+\vec{A}•\frac{\partial \vec{x}}{\partial x_j}$$