Proving differentiability and finding the matrix D : $DF|_0$

Question

The question: Given a function $F :R^n \to R^m$ such that there exists $c>0$ such that $||F(x)||\leq c||x^2||$, Prove that F is differentiable at $x=0$ and find the matrix $DF|_0$ My try: we know that $$0\leq ||F'(x)||\leq 2c||x||$$ so $$\lim \limits _{h \to 0} \frac{F(0+h)-F(0)}{h} \leq 2c||0|| = 0$$ and because $$0\leq||F(0)||\leq c||0||=0 => ||F(0)||=0 , ||F(h)||\leq c||h^2||=ch^2 $$ and same for $||F'(x)|| = 0$ so $$\lim \limits _{h \to 0} \frac{F(h)-F(0)}{h} \leq \lim \limits _{h \to 0} \frac{ch^2}{h} = \lim \limits _{h \to 0} ch = 0$$ so that proves that F is indeed differentiable. But I'm not sure what the D matrix is supposed to be. Is my answer so far is okay? How do I proceed?