example (scalar/matrix derivative via the woodbury identitylet tr - where xa rnxn we now prove that - -ax- to show thisapply the woodbury matrix identity to an infinitesimal perturbationx euof xand take to obtain the following( )- - - - ( - )- - -- - - thereforeas tr ( )- tr - --tr - - -tr - ax- nowsuppose that is an all zero matrix with one in the (ij)-th position we can write- - ( ua tr tr lim -tr ux- ax- - ax- ji | xi  - ax- thereforex the following two examples specify multivariate derivatives for the important special cases of linear and quadratic functions example (gradient of linear functionlet (xax for some mxn constant matrix thenits vector/vector derivative ( is the matrix ax ( to see thislet ai denote the (ij)-th element of aso that pn = xk (xax pn = amk xk to find the ji)-th element of xf we differentiate the -th element of with respect to fi aik xk ai = in other wordsthe (ij)-th element of xf is ji the (ij)-th element of aexample (gradient and hessian of quadratic functionlet (xxax for some constant matrix thenf ( ( ) (