I'm learning about Frechet derivatives of matrices from Bhatia's Matrix Analysis.
I want to compute the derivative of the inverse function using the formula for directional derivatives. I've seen on this website and elsewhere that the Frechet derivative of the inverse functon $g(A) = A ^{-1} $ is given by
$$Dg(A)(B) = - A ^{-1} B A ^{-1}. \tag{2}$$
I want to understand how to derive this from the directional derivative forumla for computing the Frechet derivative. Paraphrasing from Bhatia's Matrix Analysis (Eq. X.54): Let $f: \mathcal Y \to \mathcal Z$ be a continuous function. The derivative of $f$ at $A$ (if it exists) is a linear map denoted by $Df(A)$ from $\mathcal Y$ to $\mathcal Z$. It's action on an arbitrary $B \in \mathcal Y$ is then $$ Df(A)(B) = \frac {\operatorname d}{\operatorname d t} \bigg\vert_{t = 0} f(A + tB). \tag{2}$$
How does one derive Eq. (1) starting from Eq. (2)? I;ve seen answers (such as this one) where the $Dg(A)(B)$ is obtained by starting from the fact that $AA ^{-1} = I$ and then differentiating both sides, but I would like to understand it from the view of directional derivatives.
$$Dg(A)(B) = \frac{d}{dt}\Bigg|_{t=0}(A+tB)^{-1} = A^{-1}\frac{d}{dt}\Bigg|_{t=0}(I+tBA^{-1})^{-1}$$
Note that writing $A^{-1}$ is legal because we assumed $A$ is an element of the domain of $g$. Provided $BA^{-1}$ has small enough norm (which is fine because we can simply choose any direction $B$ that is small enough) then we have the Neumann series expansion, which generalises the familiar power series of $1 + x + x^2 \ldots$
$$ = A^{-1}\frac{d}{dt}\Bigg|_{t=0}\sum_{n=0}^\infty (-1)^n t^n (BA^{-1})^n = -A^{-1}BA^{-1}$$