Given an exact (additive) functor $F$, i.e. an additive functor preserving exact sequences, it is not hard to show that all derived functors of $F$ vanish.
At the same time given a right exact functor (a similar argument holds for the left exact case) one can show that for every short exact sequence $$0\longrightarrow A\longrightarrow B\longrightarrow C\longrightarrow 0$$ there exists an induced long exact sequence of the form $$\cdots \longrightarrow L_1F(B)\longrightarrow L_1F(C)\longrightarrow F(A) \longrightarrow F(B)\longrightarrow F(C) \longrightarrow 0$$ and hence if the first derived functor $L_1F$ vanishes, the functor is exact.
This seems to imply that the vanishing of the first derived functor is a sufficient condition for the vanishing of all higher derived functors. Is this true? This feels like a very strong result/constraint, so I get the feeling that I am missing something.