Let $s\in [0,1]$. How to show that the map: $F(s)=(1-s)\int_{0,s}tf(t)d\lambda (t)$ is differentiable when $f\in L_2([0,1],\lambda)$?
I tried to calculate the variation $(F(s+h)-F(s)).h^{-1}$ but without result. Also tried Lebesgue theorem of differentiation under integral, but not better. Please help!