I'm trying to find a expression for the matrix derivative with respect to the pseudo-inverse of a matrix. So, i have some function $f(A)$ of a matrix $A$, which is singular. If it weren't I could use that $$ \frac{df(A)}{dA^{-1}} = -A^{-1}\frac{df(A)}{dA}A^{-1}, $$ but I can't right? So does anyone know where I could find a pseudo-inverse version of this? So basically I want an expression for $\frac{df(A)}{dA^+}$, and yes I reckon it won't be as cleas and simple as the one above. Also, does anyone know where I could find pseudo-inverse generalizations of all those classic matrix inversion lemmas?

Thanks in advance for any comments!