Let $f:L(E)\to L(E),$ $u\mapsto f(u)=u^3=u\circ u\circ u.$ Then, $f$ is differentiable and compute $f'(u)$ Let $$f:L(E)\to L(E)$$
$$u\mapsto f(u)=u^3=u\circ u\circ u.$$
I'm interested in proving that $f$ is differentiable and computing $f'(u).$
Here is what I've done:
$$f(u+h)-f(u)=(u+h)^3-u^3$$
$$=3 u^2 h+3uh^2+h^3$$
$$=L(u) h+3uh^2+h^3$$
Then,
$$||\varepsilon (h)||=\frac{||3uh^2+h^3||}{||h||}$$
My question: I'm I on track? If yes, how do I show that $||\varepsilon (h)||\to 0$ as $h\to 0$? If no, can anyone show me a reference or a proof?
 A: Our OP Mike seems to have grasped the essential idea, but it must be remembered that, since $L(E)$ does not form a commutative algebra, we must take care to preserve the order of products when expanding expressions such as $(u + w)^n$; this is illustrated below for the function
$f(u) = u^3, \tag 1$
we have
$f(u + h) = (u + h)^3 \tag 2$
for any $h \in L(E); \tag 3$
we may expand the right-hand side as follows:
$f(u + h) = (u + h)^3 = (u + h)(u + h)(u + h) = (u + h)(u^2 + uh + hu + h^2)$
$= u^3 + u^2 h + uhu + uh^2 + hu^2 + huh + h^2 u + h^3; \tag 4$
thus
$f(u + h) - f(u) = u^2 h + uhu + uh^2 + hu^2 + huh + h^2 u + h^3$
$= u^2 h + uhu + hu^2 + uh^2 + huh  + h^2 u + h^3; \tag 5$
we note that each of the first three terms on the right of (5) is linear in $h$; thus we set, 
$D_u f(h) = u^2 h + uhu + hu^2; \tag 6$
as for the last four terms, we find
$\Vert uh^2 + huh + h^2 u + h^3 \Vert \le \Vert uh^2 \Vert + \Vert huh \Vert + \Vert h^2 u \Vert  + \Vert h^3 \Vert$
$\le 3\Vert u \Vert \Vert h \Vert^2 + \Vert h \Vert^3 = (3\Vert u \Vert + \Vert h \Vert) \Vert h \Vert^2 ; \tag 7$
we may assemble (5), (6) and (7) together to obtain
$\Vert f(u + h) - f(u) - D_u f(h) \Vert \le (3\Vert u \Vert + \Vert h \Vert) \Vert h \Vert^2, \tag 8$
whence
$\dfrac{\Vert f(u + h) - f(u) - D_u f(h) \Vert}{\Vert h \Vert} \le (3\Vert u \Vert + \Vert h \Vert) \Vert h \Vert; \tag 9$
therefore,
$\lim_{\Vert h \Vert \to 0} \dfrac{\Vert f(u + h) - f(u) - D_u f(h) \Vert}{\Vert h \Vert} = 0, \tag{10}$
which shows that our expression
$D_u f(h) = u^2 h + uhu + hu^2 \tag{11}$
itself a linear mapping $L(E) \to L(E)$, is the derivative of $f(u) = u^3$.
The above also shows how the error term $(3 \Vert u \Vert + \Vert h \Vert) \Vert h \Vert^2$ may be handled.
This result may be extended to any $g(u) = u^k$; for $D_ug(h)$ we obtain a sum of terms of the form $u^ihu^{k - i - 1}$ for $0 \le i \le k - 1$.
