Is there a simple numerical procedure for obtaining the derivative (with respect to $x$) of the pseudo-inverse of a matrix $A(x)$, without approximations (except for the usual floating-point limitations)? The matrix $\frac{\mathrm{d}}{\mathrm{d}x}A(x)$ is supposed to be known.

In other words, are there analytical formulas that could be numerically evaluated so as to obtain the derivative of the pseudo-inverse? or, what formula would generalize $$ \frac{\mathrm{d}}{\mathrm{d}x}A^{-1}(x) = -A^{-1}(x) \left(\frac{\mathrm{d}}{\mathrm{d}x}A(x)\right) A^{-1}(x) $$ for the pseudo-inverse?

I would be happy if this were possible, as this would allow my uncertainty calculation programming package to precisely calculate uncertainties on the pseudo-inverse of matrices whose elements have uncertainties (currently, a numerical differentiation is performed, which may yield imprecise results in some cases).

Any idea would be much appreciated!