Consider the inversion function $f:GL\_n( \mathbb{R}) \rightarrow GL\_n (\mathbb{R})$ , $f(X)=X^{-1}.$
Where $GL\_n( \mathbb{R})$ denotes the set of invertible $ n \times n$ matrices over the reals.

The question wants me to show that it is a differentiable function and then to calculate its derivative. It says to think of the set as a subset of $\mathbb{R} ^{n^{2}}$. 

I know that if the partials exist and are continuous then it is differentiable, I can't calculate the partials explicitly though since it seems too difficult, just thinking about it I know if I were to change 1 entry in the matrix keeping all others constant (this is how I interpret partial derivative of this function, is this correct?), I could find a neighbourhood around that entry such that the matrix is still invertible (since $det:\mathbb{R}^{n \times n} \rightarrow \mathbb{R}$ is continuous? - this has been shown in my lecture notes) Is this the correct way to go about it? I have no solutions available to me so just seeking some clarification on here to make sure my understanding isn't completely all wrong, thanks!