Derivatives of Approximate Matrix inverses

Question

I have a question concerning the derivatives of approximate matrix inverses. I have a system, $$Ax = b$$ which I solve approximately (and with an iterative method) with: $$\Delta x = x - \tilde{A}^{-1}b$$ I would like to take the derivative of this process, i.e. find $$\frac{d\Delta x}{dD} = \frac{dx}{dD} - \frac{d\tilde{A}^{-1}}{dD}b - \tilde{A}^{-1}\frac{db}{dD}$$ I know that if I had an ideal inverse, my expression would be without tildes as I'd have the exact result: $$\frac{d\Delta x}{dD} = \frac{dx}{dD} - {A}^{-1}\frac{dA}{dD}{A}^{-1}b - \tilde{A}^{-1}\frac{db}{dD}$$ But for this, I cannot start with the typical assumption that $$AA^{-1} = I$$ and instead have $$A\tilde{A}^{-1} = C \neq I$$ Does anyone have any ideas for resources or techniques for such problems? Thank you.