Consider a map $f: \mathbb{R}^{n} \rightarrow \mathbb{R}^{m}$.

Assume $f$ is differentiable at $x$ and let $D_{x} f$ denote the derivative of $f$ at $x$. Show that

$$D_{x} f(v)=\lim _{t \rightarrow 0} \frac{f(x+t v)-f(x)}{t}$$

for any $v \in \mathbb{R}^{n}$.

Assume now that $f$ is such that for some fixed $x$ and for every $v \in \mathbb{R}^{n}$ the limit

$$\lim _{t \rightarrow 0} \frac{f(x+t v)-f(x)}{t}$$

exists. Is it true that $f$ is differentiable at $x ?$ Justify your answer.

Let $M_{k}$ denote the set of all $k \times k$ real matrices which is identified with $\mathbb{R}^{k^{2}}$. Consider the function $f: M_{k} \rightarrow M_{k}$ given by $f(A)=A^{3}$. Explain why $f$ is differentiable. Show that the derivative of $f$ at the matrix $A$ is given by

$$D_{A} f(H)=H A^{2}+A H A+A^{2} H$$

for any matrix $H \in M_{k}$. State carefully the inverse function theorem and use it to prove that there exist open sets $U$ and $V$ containing the identity matrix such that given $B \in V$ there exists a unique $A \in U$ such that $A^{3}=B$.