Show that $\forall u\in L(h)$, $\|Df(u)-3Id\|\leq 6\|u-Id\|+3\|u-Id\|^2.$

Question

Let $E$ be a banach space and let $f:L(E)\to L(E)$ defined by: $f(u)=u^{3},$ $u$ in the open ball $B(Id,\frac 1 3).$

1. Show that $f$ is $C^1$ and find its differential $Df(u)(h)$ , $ \forall u,h \in L(E).$
2. Show that $\forall u\in L(h)$, $\|Df(u)-3Id\|\leq 6\|u-Id\|+3\|u-Id\|^2.$

my work:
For the question $1$ i showed that $f$ is $C^1$ and $Df(u)(h)= u^2h + uhu + hu^2$.
For $2$ we have that: $$\|Df(u)(h)-3h\|=\|u^2h + uhu + hu^2-3h\|=\|u^2h-h + uhu-h + hu^2-h\| \leq 2\|h\|\|u^2-Id\|+\|uhu-h\|.$$ I'm stuck here anyone can help me please.