Proving $\Phi$ is differentiable and find $D \Phi$

Question

Let $E = \{ f : [0,1] \to \mathbb{R} : f(0)=0 , f $ is continuously differential$ \}$

With $||f|| = \sup_{t \in [0,1]} |f'(t)|$ and Let $\Phi(f)(t) = \int \limits_{0}^{t} f(u)du$ with $||f|| = \sup_{t \in [0,1]} |f(t)|$,

prove that $\Phi$ is differentiable and compute $D\Phi$ ?

I need hint make sense of the question, like the limits is over real numbers or function that there norm tends to zero.

Answer 1

Let $a,b\in\mathbb{R}$, $\Phi(af+bg)=a\Phi(f)+b\Phi(g)$. This implies that $\Phi$ il linear and equal to its derivative.