Let $A \subset \mathbb{R}^{n}$ be an open subset. State what it means for a function $f: A \rightarrow \mathbb{R}^{m}$ to be differentiable at a point $p \in A$, and define its derivative $D f(p)$.

State and prove the chain rule for the derivative of $g \circ f$, where $g: \mathbb{R}^{m} \rightarrow \mathbb{R}^{r}$ is a differentiable function.

Let $M=M_{n}(\mathbb{R})$ be the vector space of $n \times n$ real-valued matrices, and $V \subset M$ the open subset consisting of all invertible ones. Let $f: V \rightarrow V$ be given by $f(A)=A^{-1}$.

(a) Show that $f$ is differentiable at the identity matrix, and calculate its derivative.

(b) For $C \in V$, let $l_{C}, r_{C}: M \rightarrow M$ be given by $l_{C}(A)=C A$ and $r_{C}(A)=A C$. Show that $r_{C} \circ f \circ l_{C}=f$ on $V$. Hence or otherwise, show that $f$ is differentiable at any point of $V$, and calculate $D f(C)(h)$ for $h \in M$.