I have been trying to find an analytical expression for the following:

$\frac{\partial {X^{+}}}{\partial {X}}$

In my case, $X$ has a constant rank. I've found the formula for differentiating a pseudoinverse in Golub's paper (equation 4.12):

$$ \frac{\mathrm d}{\mathrm d x} A^+(x) = -A^+ \left( \frac{\mathrm d}{\mathrm d x} A \right) A^+ +A^+ A{^+}^T \left( \frac{\mathrm d}{\mathrm d x} A^T \right) (1-A A^+) + (1-A^+ A) \left( \frac{\mathrm d}{\mathrm d x} A^T \right) A{^+}^T A^+ $$

but I can't see how to input the original matrix.