A vector differentiation operator is defined as

which can be applied to any scalar function to find its derivative with respect to :

Vector differentiation has the following properties:

To prove the third one, consider the th element of the vector:

Putting all elements in vector form, we have the above. If is symmetric, then we have

In particular, when , we have

You can compare these results with the familiar derivatives in the scalar case:

A matrix differentiation operator is defined as

which can be applied to any scalar function :

Specifically, consider
, where and
are and constant vectors, respectively, and
is an matrix. Then we have: