What is derivative of a vector respect to another vector?

I am confused. I never seen derivative of a vector respect to another vector. When I go on the web, the article just show divergence, curl, gradient etc. But not derivative of a vector respect to another vector?

For example what is $\displaystyle \frac{d(\vec{x}-\vec{x_0})^2}{d \vec{x}} ?$ For $\displaystyle \vec{x_0}$ is a constant vector.

The book seems to imply:

$\displaystyle \frac{d[(\vec{x}-\vec{x_0})^2]}{d \vec{x}} = 2(\vec{x}-\vec{x_0}) \frac{d \vec{x}}{d \vec{x}} = 2(\vec{x}-\vec{x_0}) $

I guess I don't know how to do a derivative like this. Can anyone help?