Vtyjkyuk

I have been faced with this problem. Let $$f:Eto BbbR$$
$$pmapsto f(p)=int_0^1p^3(t)dt.$$
I want to prove that $f$ is differentiable and also compute $f'(u).$ $E=R_n[x]$ is provided with the following norm $Vert pVert=sup|p(t)|,;;forall;tin [0,1]$

Here is what I've done:

$f(p) = int_0^1p^3(t)dt, tag 1$

we have

$f(p + h) = int_0^1(p + h)^3(t)dt tag 2$

for any $h in E; tag 3$

we may expand the right-hand side as follows:

$(p + h)^3 = (p + h)(p + h)(p + h) = (p + h)(p^2 + ph + hp + h^2)$
$= p^3 + p^2 h + php + ph^2 + hp^2 + hph + h^2 p + h^3; tag 4$

thus

$$f(u + h) - f(u) = int_0^1(p^2 h + php + ph^2 + hp^2 + hph + h^2 p + h^3)(t)dt$$
$$= int_0^1(p^2 h + php + hp^2 + ph^2 + hph + h^2 p + h^3)(t)dt; tag 5$$

$L(u)(h) = int_0^1(p^2 h + php + hp^2)(t)dt; tag 6$

and

$$Vert hVert varepsilon(h)= int_0^1( ph^2 + hph + h^2 p + h^3)(t)dt $$

$$ Vert varepsilon(h)Vert= fracVert int_0^1( ph^2 + hph + h^2 p + h^3)(t)dt VertVert hVert. tag 7$$

My question is: How do I force (7) to $0$? Please, can anyone help out?

Use
$$left|int_a^b u,mathrm dtright|le (b-a)|u|$$
$$|u+v|le|u+v|$$
and
$$|uv|le|u||v|$$