19.i.3 Generalized derivatives

\( \underline{\text { condition: }} \) Show that the following mapping is differentiable and find its strong derivative: \[ \begin{array}{l} f: C[0,1] \rightarrow C[0,1], \quad f(x(t))=\int_{0}^{t} k(x(s)) d s \\ \left(k(t) \in C^{1}[0,1]\right) . \end{array}\]