I have managed to show that for $A \in \text{GL}(n, \mathbb{R})$ the function $f: \text{Mat}(n, \mathbb{R}) \rightarrow \mathbb{R}$ given by $f(X) = \text{det}X$ is differentiable at A and that $Df(A)(H) = \text{Trace}((\text{adj}A) H)$ where $H \in \text{Mat}(n, \mathbb{R})$.
Now I have to proceed by showing that this result holds for any $A \in \text{Mat}(n, \mathbb{R})$. I have proven that $\text{GL}(n, \mathbb{R})$ is dense in $\text{Mat}(\mathbb{R}^n$). I'm pretty confident that we need this in order to prove the question (intuitively), but I do not know how to proceed from here.
Level: undergrad course in multidimensional real analysis.
Thanks in advance!