Q:

Derivatives of inverse matrix

In the matrix cookbook, Eq.  59, the formula:
$$
\partial_x Y^{-1}=-Y^{-1}(\partial_x Y) Y^{-1}
$$
is provided without proof. 
How to prove this formula? 

A:

The identity
$$
YY^{-1}=I
$$
implies that (chain rule)
$$
\partial_x YY^{-1} + Y\partial_x Y^{-1}=0,
$$
so that 
$$ \partial_x Y^{-1} = -Y^{-1}\partial_x YY^{-1}.  $$

