How to compute (directly from the definition) the derivative of matrix-valued function $M^{-1}(t)$ with respect to $t$ and recover the standard result $-M^{-1}(t)\frac{dM}{dt}M^{-1}(t)$?
A similar question has been asked on this site before involving this computation several times before, but without the restriction the proof come directly from the definition. In this case, I know how to give a proof using the fact that $M^{-1}(t)M(t) = I$, and applying the product rule. However, I would like to give an argument directly from the definition if possible.
A few hours of tinkering have led me nowhere fast – the issue is that the known formula is in terms of the derivative of $M$, and all methods I know of relating $M$ to $M^{-1}$, such as through the adjugate formula seem to be too ugly to recover the formula in question. Either I'm missing something, or this problem is difficult without the slick product-rule approach.
Since this is homework (of course, why else would such an arbitrary restriction be imposed on an otherwise fine argument?), a full solution is probably not necessary.
Best Answer
$$\left( \mathrm M (t + \mathrm d t) \right)^{-1} = \left( \mathrm M (t) + \dot{\mathrm M} (t) \,\mathrm d t \right)^{-1} = \cdots = \mathrm M^{-1} (t) \color{blue}{- \mathrm M^{-1} (t) \, \dot{\mathrm M} (t) \, \mathrm M^{-1} (t)} \,\mathrm d t$$