The Wikipedia article on Jacobi's formula (which gives the differential of the determinant function) contains two proofs, the second of which begins with a lemma claiming
$\det'(I)=\operatorname{tr}$, where $\det'$ is the differential of $\operatorname{det}$.
The article does not explicitly say "total differential," but I think it is reasonable to assume that this is intended, especially because it is true and one version of Jacobi's formula (the cleanest, in my opinion) gives the total differential of $\det$ as $\det'(T)(H)=\operatorname{tr}[\operatorname{adj}(T)H]$.
But the proof proceeds to show only that the directional derivative (or "Gateaux derivative") of $\det$ is as claimed. Even though the resulting operator $\operatorname{tr}$ is linear and continuous, this is not sufficient to guarantee that the total derivative exists, as I learned here.
Am I correct that Wikipedia's proof is flawed, and if so, how can it be fixed? What I'd ideally like is a general theorem saying that if the directional derivative (in all directions) is given by a continuous linear operator $A$, and some condition $X$ is satisfied, then the total derivative exists and is equal to $A$ (and then a proof that $X$ is satisfied in the given situation).
Edit: I should add that I'm hoping to avoid something that explicitly references coordinates/components. For me, the appeal of the cited proof in the above article is its coordinate-free style. As PhoemueX points out in the comments, we could argue that, because $\det$ in coordinates is a polynomial, it has continuous partial derivatives, and hence its total derivative exists. But I'd prefer something more abstract that doesn't rely on coordinates, i.e., in the style of the original article if possible.