= Solution
For $x>-1$, interpret the endpoint $-1$ by a one-sided limit. Part (iii) shows that $g$ extends continuously there with value zero, so both nested <integrals> converge. Let $F(x)$ denote the nested expression inside the <derivative>. The <fundamental theorem of calculus>, applied twice with moving lower endpoints, gives
$$
F'(x)=-\int_x^{-1}g(t)\,dt,\qquad F''(x)=g(x),\qquad F^{(3)}(x)=g'(x).
$$
By part (iii), $g'(x)=1$, hence
$$
\boxed{F^{(3)}(x)=1\qquad(x>-1).}
$$
A direct check is $\int_s^{-1}(t+1)\,dt=-(s+1)^2/2$, so $F(x)=(x+1)^3/6$, with the same third <derivative>.
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