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 Goulob'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. I've tried

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

but it doesn't seem to give the right answers. What should $\frac{\mathrm d A^T}{\mathrm d A}$ and $\frac{\mathrm d A}{\mathrm d A}$ evaluate to?