4)$\displaystyle F(x)=sin(\int_0^x sin(\int_0^y sin^3(t)dt)dy)$. I am not able to do this one. But I have started something : I wrote $\displaystyle F$ as a composition of 3 functions, that is $\displaystyle f \circ g \circ h$.

$\displaystyle f(g)=sin(g)$.

$\displaystyle g(h)=\int_0^x sin(h)dy$.

$\displaystyle h(y)=\int_0^y sin^3(t)dt$.

$\displaystyle f'(g)=cos(g)$

$\displaystyle g'(h)=$... I couldn't find it. I guess I'm already wrong writing $\displaystyle g(h)$. All those variables confuse me a lot!

$\displaystyle h'(y)=sin^3(y)$.

I wanted to use the chain rule for this one. $\displaystyle F'(x)$ would be $\displaystyle (f' \circ g *g' \circ h * h')(x)$. I'm not even sure if this is right.